Interference management and decentralized channel access schemes in hotspot-aided cellular networks

ABSTRACT

A system and method are provided wherein one or more femtocell base stations are deployed within a range of a cellular base station and utilize substantially the same frequency band as the cellular base station. Each femtocell base station may be configured to employ one or more interference avoidance techniques such that coexistence between the cellular and the corresponding femtocell base station is enabled. The interference avoidance techniques employed may include use of randomized time or frequency hopping; randomly selecting a predetermined number, or identifying one or more unutilized, frequency subchannels for signal transmission; using two or more transmit and two or more receive antennas; nulling one or more transmissions in a direction of a nearby cellular base station user; handing off at least one cellular user to one of the femtocell base stations and vice versa; and/or reducing the transmission power of at least one femtocell base station.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims benefit of and priority to U.S. ProvisionalPatent Application Ser. No. 61/116,531, filed Nov. 20, 2008, which isfully incorporated herein by reference, and made a part hereof.

FIELD

Embodiments of the present methods and system are generally directedtoward cellular wireless networks. More specifically, embodiments of theinvention are directed toward two-tier cellular networks consisting of atypical macrocell overlaid or underlaid with supplemental infrastructurethat has a shorter range than the macrocell base station, and, inparticular, toward reducing interference and efficiently accommodatingmultiple users in these two-tier cellular networks.

BACKGROUND

Wireless communication systems are constantly being pushed toaccommodate the conflicting goals of higher data rates per user, andimproved coverage area. A primary way to meet both of these desirablegoals is by deploying more infrastructures (connected into the wiredbackbone, and eventually the PSTN and/or Internet). This infrastructureis generally quite expensive, in particular the base stations thatcomprise most of the infrastructure in cellular networks.

Two-tier networks, comprising a conventional cellular network overlaidwith shorter range hotspots (e.g. femtocells, distributed antennas, orwired relays), offer an economically viable way to improve cellularsystem capacity. Femtocells are recently attracting interest forincreasing overall system capacity and coverage, particularly forsubscribers who are at home or in another common location. Femtocellsare small virtual base stations that are usually deployed by the enduser (perhaps with subsidy or logistical help from the serviceprovider). This can result in a win-win: the subscriber gets high speed,reliable access at their most common locations (many subscriberscurrently complain about their service experience at home), and theservice provider unloads considerable traffic off their expensivelarge-scale network. Because this results in two spatially overlaidnetworks (Base Stations being tier 1 and femtocells being tier 2), thecomposite network is often referred to as a “two-tier” network.

Femtocells, also known as home base stations (BTSs) or access point basestations, can connect to a service provider's network via a broadbandbackhaul connection (such as digital subscriber lines (DSL), cable, oreven a radio link). Femtocells can allow service providers to extendservice coverage indoors, where access would otherwise be limited orunavailable. Femtocells can incorporate the functionality of a typicalbase station while allowing for a simpler, self contained deployment.

However, interference between femtocells and macrocells in such networkscan be a capacity-limiting factor if the femtocells and macrocells sharethe same spectrum. The cross-tier interference between macrocells andfemtocells can suffocate the capacity due to the near-far problem, so inpractice hotspots would typically want to use a different frequencychannel than the potentially nearby high-power macrocell users.Centralized or coordinated frequency planning, which is difficult andinefficient even in conventional cellular networks, is even moredifficult in a two-tier network. Alternatively, using expensive wirelessspectrum to coordinate between the cellular network and hotspots may beself-defeating as it undermines the principle argument—low capital andoperating expenditures—for deploying femtocells in the first place.

On the other hand, femtocells and other types of supplementalinfrastructure are likely to be deployed either randomly by users of thecellular network, or on an as-needed basis by the service provider.Allocating dedicated spectrum just for these devices and the mobilestations (MSs) interacting with them is highly undesirable since theymay be sparse in many areas, rolled out slowly, and the demands on theavailable spectrum are intense, which is largely what motivates thesehotspots in the first place. Therefore, methods and techniques thatachieve frequency reuse between the two tiers are highly desirable.

In a shared spectrum two-tier network, near-far effects arising fromcross-tier interference can create problems due to conventional signalstrength based power control and can be particularly severe in a “closedaccess” deployment, where a femtocell allows only licensed subscribersto communicate with it. The worst-case scenario arises either when ahigh powered macrocell user on the cell edge causes interference tonearby femtocells, or when cell interior femtocell users [resp.femtocell BSs] create unacceptable interference to the macrocell basestation [resp. nearby cellular users].

SUMMARY

Wireless operators are in the process of augmenting the macrocellularnetwork with supplemental infrastructure such as microcells, distributedantennas and relays. An alternative with lower upfront costs is toimprove indoor coverage and capacity using the concept of end-consumerinstalled femtocells or home base stations. A femtocell can serve as alow power, short range data access point that provides high qualityin-building coverage to home users, while backhauling their traffic overthe IP network. The femtocell radio range can be much smaller than themacrocell radius. Users transmitting to femtocells can experiencesuperior indoor signal reception and can lower their transmit power,thus prolonging battery life. The implication is that femtocells canprovide higher spatial reuse and cause less cochannel interference (CCI)to other users.

As noted above, however, in a shared spectrum two-tier network, near-fareffects arising from cross-tier interference can create problems due toconventional signal strength based power control and can be particularlysevere in a “closed access” deployment, where a femtocell allows onlylicensed subscribers to communicate with it. Embodiments of the presentinvention provide solutions to tackle cross-tier interference in such anetwork—with two sets of users with dramatically uneven channel powersin their BSs—including: (1) enabling femtocell users to decrease theirtarget data rates or employ adaptive power strategies such as decreasingtheir target data rate in the presence of a nearby cellular user, (2)employing interference avoidance through “time-hopping” (or frequencyhopping) for both macrocell and femtocell users in order to “avoid”transmitting in the same time (frequency) interval, (3) requiringfemtocells to perform “open-access” through vertical handoffs byservicing outdoor cellular users with poor channel powers to theircentral macrocell, (4) eliminating cross-tier interference by assigningmacrocell and femtocell users with orthogonal frequency resources usinga spectrum partitioning strategy, (5) providing multiple transmitantennas at femtocells and utilizing the extra spatial degrees offreedom to eliminate interference to cellular users (through beamformingtechniques) and tackling cross-tier interference from the macrocell,and/or (5) utilizing techniques for handoffs (in open access) andadaptive power control at femtocells (in closed access) when multipleantennas are used at femtocells.

In particular, according to one aspect, a system is provided thatcomprises one or more femtocell base stations deployed within a range ofa cellular base station, wherein the cellular base station utilizes afrequency band and respective femtocell base stations utilize afrequency band that is substantially the same as the cellular basestation frequency band. In one embodiment, each femtocell base stationis configured to employ one or more interference avoidance techniquessuch that coexistence between the cellular base station and thecorresponding femtocell base station is enabled. According toembodiments described herein, the interference avoidance techniquesemployed may include use of randomized time or frequency hopping whentransmitting signals to and from one or more femtocell users, randomlyselecting a predetermined number of available frequency subchannels fortransmitting signals to and from the femtocell users, identifying one ormore unutilized frequency subchannels for transmitting signals to andfrom the femtocell users, using two or more transmit and two or morereceive antennas, nulling one or more transmissions in a direction of anearby cellular base station user, handing off at least one userassociated with the cellular base station to one of the femtocell basestations and vice versa, and/or reducing the transmission power of atleast one of the femtocell base stations.

According to another aspect, a method of operating one or more femtocellbase stations in conjunction with a cellular base station is provided.In one embodiment, the method may include: (1) establishing one or morefemtocell base stations within a range of a cellular base station,wherein the cellular base station utilizes a frequency band and whereinrespective femtocell base stations utilize a frequency band that issubstantially the same as the cellular base station frequency band; and(2) employing, by respective femtocell base stations, one or moreinterference avoidance techniques such that coexistence between thecellular base station and the corresponding femtocell base station isenabled. According to embodiments described herein the interferenceavoidance techniques may include any one or more of the techniquesmentioned above and described in more detail below.

Additional advantages will be set forth in part in the description whichfollows or may be learned by practice. The advantages will be realizedand attained by means of the elements and combinations particularlypointed out in the appended claims. It is to be understood that both theforegoing general description and the following detailed description areexamples and explanatory only and are not restrictive, as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, not drawn to scale, which are incorporated inand constitute a part of this specification, illustrate embodiments andtogether with the description, serve to explain the principles of themethods and systems:

FIG. 1 illustrates an example of a cellular system with macrocells andfemtocells;

FIG. 2 illustrates potential interference between macrocell andfemtocell transmissions;

FIG. 3 illustrates an exemplary operating environment for performing themethods of embodiments of the present invention;

FIG. 4 illustrates a two-tier femtocell network with DS-CDMAtransmissions;

FIG. 5 illustrates a table of system parameters;

FIG. 6 illustrates intra-tier and cross-tier co-channel interference(CCI) at each tier;

FIG. 7 illustrates a comparison of joint and independent hoppingprotocols at a femtocell base station with antenna sectoring;

FIG. 8 illustrates outage lower bounds for interior and cornerfemtocells;

FIG. 9 illustrates macrocell outage performance with femtocellexclusion;

FIG. 10 illustrates a macrocell operating contour (OC);

FIG. 11 illustrates interior femtocell OC;

FIG. 12 illustrates network OCs for different macrocell-femtocellreceived power ratios and fixed hopping slots;

FIG. 13 illustrates network OCs with different hopping slots and cornerfemtocell reference;

FIG. 14 illustrates network OCs with different hopping slots andinterior femtocell reference;

FIG. 15 illustrates network OCs with tier selection and femtocellexclusion and interior femtocell;

FIG. 16 illustrates spatially averaged macrocell subchannel throughput;

FIG. 17 illustrates theoretical and empirical throughput per femtocellρ_(f)T_(f);

FIG. 18 illustrates femtocell area spectral efficiencies;

FIG. 19 illustrates optimal spectrum allocation ρ for varying Quality ofService (QoS) parameter η;

FIG. 20 illustrates area spectral efficiencies in a two-tier network forvarying QoS parameter η;

FIG. 21 illustrates the average network-wide throughput ρ_(f)T_(f) (1−ρ)provided by femtocells in their allocated spectrum (1−ρ);

FIG. 22 illustrates the spectrum WF meeting a target average data rateof D_(c)=0.1 Mbps for each macrocell user, given Round-Robin andProportional-Fair scheduling at the macrocell;

FIG. 23 illustrates another table of system parameters;

FIG. 24 illustrates dead zones caused by cross-tier interference in ashared spectrum two-tier network;

FIG. 25 illustrates transmission scheme at femtocell;

FIG. 26 illustrates examples showing two cellular users experiencingequal path-losses to B₀ and B₁ respectively;

FIG. 27 illustrates two antennas per femtocell and one indoor userserved per femtocell, canceling cross-tier interference requirescoordinated design of precoder F;

FIG. 28 illustrates system parameters;

FIG. 29 illustrates outage probability at femtocell as a function of itsdistance from the central macrocell; and

FIG. 30 illustrates threshold distance of cellular user at which QoSrequirement ε=0.1 is violated as a function of the number of femtocellsper cell site and the number of transmit antennas at the macrocell M.

DETAILED DESCRIPTION

Before embodiments of the present methods and systems are disclosed anddescribed, it is to be understood that the methods and systems are notlimited to specific methods, specific components, specific systems or toparticular compositions. It is also to be understood that theterminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting.

As used in the specification and the appended claims, the singular forms“a”, “an” and “the” include plural referents unless the context clearlydictates otherwise. Ranges may be expressed herein as from “about” oneparticular value, and/or to “about” another particular value. When sucha range is expressed, another embodiment includes from the oneparticular value and/or to the other particular value. Similarly, whenvalues are expressed as approximations, by use of the antecedent“about,” it will be understood that the particular value forms anotherembodiment. It will be further understood that the endpoints of each ofthe ranges are significant both in relation to the other endpoint, andindependently of the other endpoint.

“Optional” or “optionally” means that the subsequently described eventor circumstance may or may not occur, and that the description includesinstances where said event or circumstance occurs and instances where itdoes not.

“Exemplary” means “an example of” and is not intended to convey ameaning of an ideal or preferred embodiment. Further, the phrase “suchas” as used herein is not intended to be restrictive in any sense, butis merely explanatory and is used to indicate that the recited items arejust examples of what is covered by that provision. “Such as” is notused in a restrictive sense, but for explanatory purposes.

Disclosed are components that can be used to perform the disclosedmethods and systems. These and other components are disclosed herein,and it is understood that when combinations, subsets, interactions,groups, etc. of these components are disclosed that while specificreference of each various individual and collective combinations andpermutation of these may not be explicitly disclosed, each isspecifically contemplated and described herein, for all methods andsystems. This applies to all aspects of this application including, butnot limited to, steps in disclosed methods. Thus, if there are a varietyof additional steps that can be performed it is understood that each ofthese additional steps can be performed with any specific embodiment orcombination of embodiments of the disclosed methods.

Embodiments of the present methods and systems may be understood morereadily by reference to the following detailed description ofembodiments and the examples included therein and to the figures andtheir previous and following description.

Overview

In general, embodiments of the present invention provide an improvementover the known prior art by providing, among other things, severaltechniques for reducing or managing the interference associated withtwo-tier cellular networks consisting of a central macrocell and one ormore femtocells, or other similar supplemental infrastructure. Further,embodiments of the present invention are practically implementable intwo-tier networks irrespective of whether they employ a “closed-access”scheme (i.e., wherein each femtocell provides service to a few licensedprivileged in-home users), or an “open-access” scheme (i.e., wherein acellular user experiencing a poor channel to its macrocell may performvertical handoff and receive service from a nearby femtocell).

Where Underlay operation (universal frequency reuse) is employed, in oneembodiment of the invention, co-channel interference between the outdoorcellular macrocell transmissions and indoor femtocell transmissions maybe reduced by employing directional antennas at each femtocell.Directional femtocell antennas may reduce the geographic region withinwhich a femtocell user “sees” cross-tier interference from cellularusers. In another embodiment of the invention, a randomized time-hopped(TH) or frequency hopped (FH) transmission strategy in each tier can beused to reduce interference. Since TH uses randomized channel access,the probability that a cellular user and a femtocell user accessing acommon frequency resource at the same time instant may be significantlyreduced. Additionally, a TH strategy can be augmented with additionalinterference suppression in the form of code-division-multiple-access(CDMA) to further reduce co-channel interference. Similarly, a FH-CDMAstrategy may divide a given region of spectrum into multiplesubchannels. Randomized frequency sub channel access by cellular [resp.indoor femtocell] users may significantly reduce the probability ofcollision—and hence cross-tier interference caused by simultaneouscellular and femtocell transmissions—in the same subchannel.

With Overlay operation (when macrocell and femtocells operate inseparate regions of the spectrum) in a two-tier network with frequencydivision multiple access (FDMA) transmission, in one embodiment of theinvention, spectrum allocation or partitioning can be employed. Spectrumpartitioning can provide an efficient method for allocating spectrumbetween tiers by taking into account the loading of users at each tier.As users move between the macrocell (cellular) network to the femtocellnetwork and vice versa, the methods and systems can dynamically loadbalance spectrum by varying allocated spectrum between the macrocell andfemtocell networks. In another embodiment of the invention,Frequency-ALOHA (distinguishable from the conventional ALOHA, whichrefers to transmissions conducted in the time domain) can be employed.By taking into account the mean number of femtocells at each cell-site,the terrestrial propagation parameters, such as path-loss exponents andwall partition losses, and the characteristics of wireless channels, theyield in the optimal portion of spectrum which can be accessed by eachfemtocell in order to maximize the spatial reuse of spectrum amongco-channel femtocells can be determined. In yet another embodiment ofthe invention, carrier sensed spectrum access can be employed to enablea femtocell to opportunistically detect and transmit over unutilizedsubchannels in order to avoid interfering with nearby femtocells.

Yet another embodiment of the invention proposes use of multipleantennas—also called Multiple-Input-Multiple-Output (MIMO)transmission—at the central macrocell/femtocells. MIMO can provide extradegrees of freedom (spatial diversity) to combat cross-tier interferencein a two-tier network with universal frequency reuse. Specifically,using multiple antennas may enable a femtocell to (a) eliminatecross-tier interference to nearby cellular users in a closed-accesssystem by nulling their beams in the direction of a nearby cellularuser, (b) handoff nearby cellular users and boost their signal strengththrough array processing, (c) utilize the extra spatial degrees offreedom to eliminate interference to cellular users (through beamformingtechniques) and tackle cross-tier interference from the macrocell and/or(d) use techniques for handoffs (in open access) and adaptive powercontrol at femtocells (in closed access).

Overall System & Femtocell Base Station

FIG. 1 is a simplified illustration of a two-tier cellular systemaccording to one embodiment of the present invention. As shown, amacrocell 201 is a cell in a cellular network that provides radiocoverage served by a power cellular base station 201 (tower) to one ormore wireless devices 210, 211 (e.g., cellular telephones, personaldigital assistants (PDAs), laptops, etc.). Each cell within the cellularnetwork typically employs a standardized method for each distributedstation to distinguish the signal emanating from its own transmitterfrom the signals received from other transmitters. Two standards knownto those of ordinary skill in the art, are frequency division multipleaccess (FDMA) and code division multiple access (CDMA). FDMA uses achannel access method in multiple-access protocols as a channelizationprotocol. FDMA can give users an individual allocation of one or severalfrequency bands, allowing them to utilize the allocated radio spectrumwithout interfering with each other. CDMA employs spread-spectrumtechnology and a special coding scheme (where each transmitter isassigned a code) that can allow multiple users to be multiplexed overthe same physical channel.

Macrocells generally provide the widest geographic area coverage.However, some locations at the boundaries of an individual macrocell'sgeographic area of a plurality of macrocells or within the boundaries(such as inside a home or business) can suffer from low signal strength.To improve coverage at these locations, femtocells 220, 230, 240 can bedeployed. Femtocells 220, 230, 240, sometimes referred to as accesspoint base stations, can be small cellular base stations designed foruse in residential or small business environments. Each femtocell mayprovide coverage to one or more wireless devices 221, 222, 231, 232,241, 242. Femtocells can connect to a service provider's network 250 viaa broadband backhaul connection 262, 263, 264 (such as DSL, cable orradio link). A femtocell can allow service providers to extend servicecoverage indoors, where access would otherwise be limited orunavailable. Femtocells can incorporate the functionality of a typicalbase station with a simpler, self contained deployment. However, becausefemtocells use the same transmission frequencies as macrocells,interference can occur.

FIG. 2 is a simplified illustration of when interference can occur. Iffemtocell 240 and macrocell 201 are using the same frequencies, as auser 211 travels to the boundary of a macrocell 201 the signal 301 frommacrocell 201 and the signal 241 from femtocell 240 can interfere witheach other, causing problems with communication with both macrocell 201and femtocell 240. Similarly, while not specifically shown, interferencemay occur from macrocell 201 at femtocell 220, which is very close tocentral macrocell 201) or 230, affecting users 221, 222, 231 and 232.

As discussed below, according to embodiments of the present invention,each femtocell may be configured to manage/reduce this type ofinterference using one or more interference avoidance techniques.Accordingly, each femtocell may comprise, among other things, aprocessor configured to perform the interference management techniquesdescribed herein and/or memory storing an application which, whenexecuted, causes the processor to perform some combination of thosetechniques.

According to one embodiment, the femtocell base station may comprise acomputer 101 like the one shown in FIG. 3. Referring to FIG. 3, oneskilled in the art will appreciate that provided herein is a functionaldescription, and that respective functions can be performed by software,hardware, or a combination of software and hardware associated with thefemtocell base station. A unit can be software, hardware, or acombination of software and hardware operably connected to a basestation. The units can comprise interference avoidance and decentralizedchannel access schemes software 106 as illustrated in FIG. 3 anddescribed below.

FIG. 3 is a block diagram illustrating an exemplary operatingenvironment for performing the disclosed method. This exemplaryoperating environment is only an example of an operating environment andis not intended to suggest any limitation as to the scope of use orfunctionality of operating environment architecture. Neither should theoperating environment be interpreted as having any dependency orrequirement relating to any one or combination of components illustratedin the exemplary operating environment.

Embodiments of the present methods and systems can be operational withnumerous other general purpose or special purpose computing systemenvironments or configurations. Examples of well known computingsystems, environments, and/or configurations that can be suitable foruse with the system and method comprise, but are not limited to,personal computers, server computers, laptop devices, and multiprocessorsystems. Additional examples comprise set top boxes, programmableconsumer electronics, network PCs, minicomputers, mainframe computers,distributed computing environments that comprise any of the abovesystems or devices, and the like.

The processing of the disclosed methods and systems can be performed bysoftware components. The disclosed system and methods can be describedin the general context of computer-executable instructions, such asprogram modules, being executed by one or more computers or otherdevices. Generally, program modules comprise computer code, routines,programs, objects, components, data structures, etc. that performparticular tasks or implement particular abstract data types. Thedisclosed methods can also be practiced in grid-based and distributedcomputing environments where tasks are performed by remote processingdevices that are linked through a communications network. In adistributed computing environment, program modules can be located inboth local and remote computer storage media including memory storagedevices.

Further, one skilled in the art will appreciate that the system andmethod disclosed herein can be implemented via a general-purposecomputing device in the form of a computer 101. The components of thecomputer 101 can comprise, but are not limited to, one or moreprocessors or processing units 103, a system memory 112, and a systembus 113 that couples various system components including the processor103 to the system memory 112. The processor 103 can containnon-identical multiple cores. Such a processor 103 is referred to as aheterogeneous multi-core processor. In the case of multiple processingunits 103, the system can utilize parallel computing.

The system bus 113 represents one or more of several possible types ofbus structures, including a memory bus or memory controller, aperipheral bus, an accelerated graphics port, and a processor or localbus using any of a variety of bus architectures. By way of example, sucharchitectures can comprise an Industry Standard Architecture (ISA) bus,a Micro Channel Architecture (MCA) bus, an Enhanced ISA (EISA) bus, aVideo Electronics Standards Association (VESA) local bus, an AcceleratedGraphics Port (AGP) bus, a Peripheral Component Interconnects (PCI) bus,a PCI-Express bus, a Personal Computer Memory Card Industry Association(PCMCIA) bus, Universal Serial Bus (USB) and the like. The bus 113, andall buses specified in this description can also be implemented over awired or wireless network connection and each of the subsystems,including the processor 103, a mass storage device 104, an operatingsystem 105, interference avoidance and decentralized channel accessschemes software 106, interference avoidance and decentralized channelaccess schemes data 107, a network adapter 108, system memory 112, anInput/Output Interface 116, a display adapter 109, a display device 111,and a human machine interface 102, can be contained within one or moreremote computing devices 114 a,b,c at physically separate locations,connected through buses of this form, in effect implementing a fullydistributed system.

The computer 101 typically comprises a variety of computer readablemedia. Exemplary readable media can be any available media that isaccessible by the computer 101 and comprises, for example and not meantto be limiting, both volatile and non-volatile media, removable andnon-removable media. The system memory 112 comprises computer readablemedia in the form of volatile memory, such as random access memory(RAM), and/or non-volatile memory, such as read only memory (ROM). Thesystem memory 112 typically contains data such as interference avoidanceand decentralized channel access schemes data 107 and/or program modulessuch as operating system 105 and interference avoidance anddecentralized channel access schemes software 106 that are immediatelyaccessible to and/or are presently operated on by the processing unit103.

In another aspect, the computer 101 can also comprise otherremovable/non-removable, volatile/non-volatile computer storage media.By way of example, FIG. 3 illustrates a mass storage device 104 whichcan provide non-volatile storage of computer code, computer readableinstructions, data structures, program modules, and other data for thecomputer 101. For example and not meant to be limiting, a mass storagedevice 104 can be a hard disk, a removable magnetic disk, a removableoptical disk, magnetic cassettes or other magnetic storage devices,flash memory cards, CD-ROM, digital versatile disks (DVD) or otheroptical storage, random access memories (RAM), read only memories (ROM),electrically erasable programmable read-only memory (EEPROM), and thelike.

Optionally, any number of program modules can be stored on the massstorage device 104, including by way of example, an operating system 105and interference avoidance and decentralized channel access schemessoftware 106. Each of the operating system 105 and interferenceavoidance and decentralized channel access schemes software 106 (or somecombination thereof) can comprise elements of the programming.Interference avoidance and decentralized channel access schemes data 107can also be stored on the mass storage device 104. Interferenceavoidance and decentralized channel access schemes data 107 can bestored in any of one or more databases known in the art. Examples ofsuch databases comprise, DB2®, Microsoft® Access, Microsoft® SQL Server,Oracle®, mySQL, PostgreSQL, and the like. The databases can becentralized or distributed across multiple systems.

In another aspect, the user can enter commands and information into thecomputer 101 via an input device (not shown). Examples of such inputdevices comprise, but are not limited to, a keyboard, pointing device(e.g., a “mouse”), a microphone, a joystick, a scanner, tactile inputdevices such as gives, and other body coverings, and the like These andother input devices can be connected to the processing unit 103 via ahuman machine interface 102 that is coupled to the system bus 113, butcan be connected by other interface and bus structures, such as aparallel port, game port, an IEEE 1394 Port (also known as a Firewireport), a serial port, or a universal serial bus (USB).

In yet another aspect, a display device 111 can also be connected to thesystem bus 113 via an interface, such as a display adapter 109. It iscontemplated that the computer 101 can have more than one displayadapter 109 and the computer 101 can have more than one display device111. For example, a display device can be a monitor, an LCD (LiquidCrystal Display), or a projector. In addition to the display device 111,other output peripheral devices can comprise components such as speakers(not shown) and a printer (not shown) which can be connected to thecomputer 101 via Input/Output Interface 116. Any step and/or result ofthe methods can be output in any form to an output device. Such outputcan be any form of visual representation, including, but not limited to,textual, graphical, animation, audio, tactile, and the like.

The computer 101 can operate in a networked environment using logicalconnections to one or more remote computing devices 114 a, b, c. By wayof example, a remote computing device can be a personal computer,portable computer, a server, a router, a network computer, a peer deviceor other common network node, and so on. Logical connections between thecomputer 101 and a remote computing device 114 a, b, c can be made via alocal area network (LAN) and a general wide area network (WAN). Suchnetwork connections can be through a network adapter 108. A networkadapter 108 can be implemented in both wired and wireless environments.Such networking environments are conventional and commonplace inoffices, enterprise-wide computer networks, intranets, and the Internet117.

For purposes of illustration, application programs and other executableprogram components such as the operating system 105 are illustratedherein as discrete blocks, although it is recognized that such programsand components reside at various times in different storage componentsof the computing device 101, and are executed by the data processor(s)of the computer. An implementation of interference avoidance anddecentralized channel access schemes software 106 can be stored on ortransmitted across some form of computer readable media. Any of thedisclosed methods can be performed by computer readable instructionsembodied on computer readable media. Computer readable media can be anyavailable media that can be accessed by a computer. By way of exampleand not meant to be limiting, computer readable media can comprise“computer storage media” and “communications media.” “Computer storagemedia” comprise volatile and non-volatile, removable and non-removablemedia implemented in any method or technology for storage of informationsuch as computer readable instructions, data structures, programmodules, or other data. Exemplary computer storage media comprises, butis not limited to, RAM, ROM, EEPROM, flash memory or other memorytechnology, CD-ROM, digital versatile disks (DVD) or other opticalstorage, magnetic cassettes, magnetic tape, magnetic disk storage orother magnetic storage devices, or any other medium which can be used tostore the desired information and which can be accessed by a computer.

The methods and systems can employ Artificial Intelligence techniquessuch as machine learning and iterative learning. Examples of suchtechniques include, but are not limited to, expert systems, case basedreasoning, Bayesian networks, behavior based AI, neural networks, fuzzysystems, evolutionary computation (e.g. genetic algorithms), swarmintelligence (e.g. ant algorithms), and hybrid intelligent systems (e.g.Expert inference rules generated through a neural network or productionrules from statistical learning).

Computer processors are often designed with multiple processor cores. Aplurality of computer programs may run on a computer and an operatingsystem on the computer can be tasked with determining which program runson which core. In one embodiment of the methods and systems amathematical system that analyzes programs to determine which core theyshould be assigned to. The mathematical system, referred to as a fuzzysystem or fuzzy logic, can be utilized to determine a desired assignmentof applications to cores.

Techniques for Reducing Interference in CDMA Cellular Network

According to embodiments of the present invention, in a two-tier networkwith universal frequency reuse, such as a two-tier network employingcode division multiple access (CDMA) transmission (e.g., IS-95,CDMA2000, UMTS, etc.), several techniques may be used to enablefemtocells to eliminate interference from conventional signal-strengthbased cellular power control. In particular, according to oneembodiment, directional antennas at the femtocells may be used. As oneof ordinary skill in the art can recognize, directional antennas referto antennas which radiate greater power in one or more directionsallowing for increased performance on transmit and receive, and reducedinterference from unwanted sources.

While using directional antennas at the macrocell base station level maybe known to those of ordinary skill in the art, this embodiment of theinvention utilizes directional antennas at the femtocell base stationlevel. Additionally, by using directional antennas at the femtocelllevel, the cross-tier cochannel interference (CCI) caused by nearbymacrocellular users which can lead to outage performance over thefemtocell uplink can be reduced.

Providing directional antennas may benefit indoor femtocell users byrestricting the region within which they experience interference fromco-channel outdoor cellular transmissions. In addition, there may bezero associated protocol overhead.

According to another embodiment, randomized time-hopping may be used. Ingeneral, in time-hopped CDMA, the CDMA duration G T_(c), where G can bethe processing gain and T_(c) can be the chip period, can be dividedinto N_(hop) hopping slots, where each user can randomly select ahopping slot for transmission and remain silent during the remainingslots.

According to embodiments of the present invention, use of randomizedtime-hopping may enable users to avoid rather than suppress mutualinterference in a decentralized manner. Since time hopping usesrandomized channel access, the probability that a cellular user and afemtocell user will access the common frequency resource as the sametime instant may be significantly reduced. According to one embodiment,time hopping may be augmented with additional interference suppressionin the form of CDMA to further reduce co-channel interference.

According to yet another embodiment, frequency hopping may be used. Inparticular, according to one embodiment, the femtocell may rapidly, andrandomly, switch carriers among many frequency channels using apseudorandom sequence known to both the transmitter and the receiver.

The following provides a more detailed description and analysis of theuse of directional antennas and randomized time hopping in a two-tiernetwork employing CDMA in order to reduce interference, in accordancewith embodiments of the present invention.

I. SYSTEM MODEL

Referring to FIG. 4, which illustrates a two-tier femtocell network withdirect sequence (DS) CDMA, according to one embodiment, H⊂R² may bedenoted as the interior of a reference hexagonal macrocell C of radiusR_(c). The tier 1 network can consist of low density macrocellular usersthat are communicating with the central base station in each cell site.The macrocellular users are distributed on R² according to a homogeneousSPPP Ωc of intensity λ_(c). The overlaid tier 2 network containing thefemtocell base stations forms a homogeneous SPPP Ω_(f) with intensityλ_(f) (the system model allows a macrocellular user to be present insidea femtocell as the governing process Ωc is homogeneous). Each femtocellhotspot includes a Poisson distributed population of activelytransmitting users with mean U_(f) in a circular coverage area of radiusR_(f), R_(f)<<R_(c) (a hard handoff may be assumed to allocatesubscribed hotspot users to a femtocell, provided they fall within itsradio range). To maximize user capacity per cell site, it may bedesirable to have λ_(f)>>λ_(c); as will be shown, cross-tier CCI at amacrocell base station limits λ_(f) for a given λ_(c). Defining |H|

2.6R_(r) ² as the area of the hexagonal region H, the mean number ofmacrocell users and femtocell base station's per cell site are given asN_(c)=λ_(c)·|H| and N_(f)=λ_(f)·|H| respectively. FIG. 5 shows anexemplary summary of system parameters and typical values for them,which are used later in numerical simulations.

According to the embodiments described herein, users in each tier employDS-CDMA with processing gain G. Uplink power control adjusts forpropagation losses and log-normal shadowing, which is standard incontemporary CDMA networks. The macrocell and femtocell receive powersare denoted as P_(r) ^(c) and P_(r) ^(f) respectively. Any power controlerrors and short-term fading effects are ignored for analyticalconvenience. This assumption has been affirmed as reasonable, especiallyin a wideband system with significant frequency diversity and robustreception (through RAKE receiver, coding and interleaving).

A. TH-CDMA and Antenna Sectoring

In one embodiment, suppose that the CDMA period T=C·T_(C) is dividedinto N_(hop) hopping slots, each of duration T/N_(hop). Macrocell usersand femtocell users (active users within a femtocell may transmit in thesame hopping slot) can independently choose to transmit over any oneslot, and remain silent over the remaining N_(hop)−1 slots. Theresulting intra-tier and cross-tier interference can be “thinned” by afactor of N_(hop). Using Time Hopping-CDMA (TH-CDMA), users in each tiercan effectively sacrifice a factor N_(hop) of their processing gain, butbenefit by thinning the interfering field by the same factor.

Referring to FIG. 6, which illustrates intra-tier and cross-tier CCI ateach tier and in which the arrows denote the CCI arising from eithertier 1 (i.e., the macrocell) or tier 2 (i.e., the femtocell), assumesectored antenna reception in both the macrocell and femtocell basestation, with antenna alignment angle θ and sector width equaling2π/N_(sec). While antenna sectoring is a common feature at the macrocellBS in practical cellular systems, embodiments of the present inventionuse sectored antennas at femtocell BS's as well. One reason is that thecross-tier CCI caused by nearby macrocellular users can lead tounacceptable outage performance over the femtocell uplink; this maymotivate the need for directed femtocell antennas. The spatial thinningeffect of TH-CDMA transmission and antenna sectoring may be derived inthe following lemma.

Lemma 1 (Spatial thinning by interference avoidance): With TH-CDMAtransmission over N_(hop) slots and antenna sectoring with N_(sec)directed base station antennas in each tier, the interfering field at agiven base station antenna sector can be mapped to the Spatial PoissonPoint Process (SPPP) Φ_(C) and Φ_(f) on

² with intensities η_(C)=λ_(C)/(N_(hop)·N_(sec)) andη_(f)=λ_(f)(1−e^(−U) ^(f) )/(N_(hop)·N_(sec)) respectively. The SPPP isa natural model arising from mobility of macrocellular users andplacement of femtocell base station in densely populated areas, and hasbeen confirmed in empirical studies and used by those of ordinary skillin the art.

The following definitions may be useful.

Definition 1: Denote

⊂

as the region within

covered by a base station antenna sector corresponding to a macrocellbase station or a femtocell base station within the reference cellsite.For example,

=

for an omnidirectional femtocell located at the corner of the referencemacrocell.

Definition 2: Denote {circumflex over (Ω)}_(C) and {circumflex over(Ω)}_(f) as the heterogeneous SPPPs composed of active macrocell andfemtocell interferers as seen at a base station antenna sector in eachtier, whose intensities are given by {circumflex over (λ)}_(C) and{circumflex over (λ)}_(f). Denote the equivalent mapped homogeneousSPPPs over

². Φ_(C) and Φ_(f) whose intensities are given by η_(C) and η_(f)respectively.

Definition 3: Denote the restriction of {circumflex over (Ω)}_(C) and{circumflex over (Ω)}_(f) to

by the SPPPs π_(C) and π_(f) respectively.

B. Channel Model & Interference

In an embodiment of the invention, the channel can be represented as acombination of path-loss and log-normal shadowing. The path-lossexponents are denoted by α (outdoor transmission) and β (indoorfemtocell transmission) while lognormal shadowing can be parameterizedby its standard deviation σ_(dB).

Through uplink power control, a macrocell user transmitting at a randomposition X with respect to the reference macrocell base station C canchoose a transmit power level P_(t) ^(c)=P_(r) ^(c)/g_(c)(|X|). Hereg_(c)(|X|) can be the attenuation function defined asg_(c)(|X|)=K_(c)(d_(0c)/|X|)^(α)Θ_(C) where 10 log₁₀Θ_(C)−N(0, σ_(dB) ²)is the log-normal shadowing from user to C, K_(c)

[c/(4πf_(c)d_(0c))]² can be a unitless constant that depends on thewavelength of the RF carrier c/f_(c) and outdoor reference distanced_(0c). Similarly, a femtocell user at a random position Y within afemtocell base station F chooses a transmit power P_(t) ^(f)=P_(r)^(f)/g_(f)(|Y|), where g_(f)(|Y|)=K_(f)(d_(0f)/|Y|)^(β)Θ_(F), 10log₁₀Θ_(f)−N(0, σ_(dB) ²) and K_(f)

[c/(4πf_(c)d_(0f))]². Here d_(0f) is the reference distance forcalculating the indoor propagation loss. Note that K_(c) and K_(f) canbe empirically determined. The interference in each tier (FIG. 6) can begrouped as:

Macrocell interference at a macrocell base station (I_(c,in),I_(c,out)):Through power control, all macro-cell users within

can be received with constant power P_(r) ^(c), so the in-cellinterference equals (N−1)·P_(r) ^(c), where N˜Poisson(N_(c)/N_(hop)). Assuch, inferring the exact statistics of out-of-cell macrocellularinterference I_(c,out) is analytically intractable. It can be assumedthat I_(c,out) can be distributed according to a scaled Gaussian pdf.Defining μ and σ² to be the empirically determined parameters of theGaussian, the pdf of I_(c,out) can be given as

${{f_{I_{c,{out}}}(y)} = \frac{2^{{- \frac{1}{2}}{{({y - \mu})}^{2}/\sigma^{2}}}}{\sqrt{2{\pi\sigma}^{2}}\left\lbrack {2 - {{erfc}\left( \frac{\mu}{\sqrt{2\sigma}} \right)}} \right\rbrack}},{where}$${{erfc}(t)}\overset{\Delta}{=}{\sqrt{\frac{2}{\pi}}{\int_{t\sqrt{2}}^{\infty}{^{{- x^{2}}/2}{{x}.}}}}$

Femtocell interference at a macrocell base station (I_(c,f)): In anembodiment of the invention, femtocell F_(i) with U_(i)˜Poisson(U_(f))users can be located at random position X_(i) with respect to referencemacrocell base station C. Inside F_(i), a randomly placed Tier 2 user jat distance Y_(j) from the femtocell base station can transmit withpower P_(i) ^(f)(j)=P_(r) ^(f)/g_(f)(Y_(j)). The interference caused atC from user j inside F_(i) can be given as:

$\begin{matrix}{{{I_{c,f}\left( {F_{i},j} \right)} = {{{P_{t}^{f}{{g_{c}\left( \left. {X_{i} + Y_{j}} \right) \right.}/{g_{f}\left( {Y_{j}} \right)}}} \approx {P_{r}^{f}{{g_{c}\left( {X_{i}} \right)}/{g_{f}\left( R_{f} \right)}}}}\mspace{104mu} = {Q_{f}{\Theta_{j,C}/\Theta_{j,F_{i}}}{{X_{i}}^{- \alpha}.{Where}}}}}{Q_{f}\overset{\Delta}{=}{P_{r}^{f}R_{f}^{\beta}{\frac{K_{c}d_{0c}^{\alpha}}{K_{f}d_{0f}^{\beta}}.}}}} & (1)\end{matrix}$

In doing so, two assumptions can be made:

Assumption 1: For small sized femtocells (R_(f)<<R_(c)), a femtocell ormacrocell BS can see CCI from other femtocells as a point source ofinterference, implying g_(c)(|X_(i)+Y_(j)|)≈g_(c)(|X_(i)|).

Assumption 2: When analyzing the interference caused by a randomfemtocell F_(i) at any other location, the U_(i) femtocell users can bemodeled as transmitting with maximum power, so thatg_(f)(|Y_(j)|)≈g_(f)(R_(f)). This can be for analytical tractability andmodeling worst-case interference. Summing equation (1) over allfemtocells over an antenna sector at a macrocell base station, thecumulative cross-tier CCI at the reference macrocell base station C canbe represented by the Poisson SNP,

$\begin{matrix}{{I_{c,f} = {\sum\limits_{F_{i} \in {\hat{\Omega}}_{f}}{Q_{f}\Psi_{i}{X_{i}}^{- \alpha}}}}{where}{\Psi_{i}\overset{\Delta}{=}{\sum\limits_{t = 1}^{U_{i}}{\Theta_{I,C}/\Theta_{I,F_{i}}}}}} & (2)\end{matrix}$

can define the cumulative shadowing gain between actively transmittingusers in femtocell F_(i) and macrocell base station C.

Neighboring femtocell interference at a femtocell base station(I_(f,f)): By a similar argument as above, in another embodiment of theinvention, the interference caused at a base station antenna sector offemtocell F_(j) from other femtocells F_(i), where i≠j, can be a PoissonSNP given by

${I_{f,f} = {\sum\limits_{F_{i} \in {\hat{\Omega}}_{f}}{Q_{f}\Psi_{i}{X_{i}}^{- \alpha}}}},$

where |X_(i)| refers to the distance between (F_(i), F_(j)) and

$\Psi_{i}\overset{\Delta}{=}{\sum\limits_{l = 1}^{U}{\Theta_{I,F_{j}}/{\Theta_{I,F_{i}}.}}}$

Interference from active users within a femtocell (I_(f,in)): In yetanother embodiment, conditioned on the femtocell containing U≧1 activelytransmitting users, the intra-tier CCI experienced by the user ofinterest arising from simultaneous transmissions within the femtocellcan equal

${\left( {U - 1} \right) \cdot P_{r}^{f}},\mspace{14mu} {{E\lbrack U\rbrack} = {\frac{U_{f}}{1 - ^{- U_{f}}}.}}$

Macrocell interference at a femtocell BS (I_(f,c)): In anotherembodiment, outage probability at a femtocell base station F_(j),located on the hexagonal axis, can be analyzed by considering the effectof in-cell macrocellular CCI. The interference I_(f,c) ^(lb) arisingfrom users in Π_(c) can form a lower bound on the cumulative tier 1 CCII_(f,c) and represented as

${{I_{f,c} \geq I_{f,c}^{lb}} = {\sum\limits_{i \in \Pi_{c}}{P_{r}^{c}{\Psi_{i}\left( \frac{X_{i}}{Y_{i}} \right)}^{\alpha}}}},$

where Ψ_(i)

Θ_(i,F) _(j) /Θ_(i,C),10 log₁₀Ψ_(i)−N(0, 2σ_(dB) ²) can be the LNshadowing term and |X_(i)|, |Y_(i)| can represent the distances ofmacrocell user i to the macrocell base station and femtocell basestation respectively. Observe that a corner femtocell can experience asignificantly higher macrocell CCI relative to an interior femtocell,therefore the cdf F_(l) _(fx) (·) may not be a stationary distribution.

II. PER TIER OUTAGE PROBABILITY

In an embodiment of the invention, to derive the operating contours(OC), an uplink outage probability constraint can be formulated in eachtier. N_(f) and N_(c) can be defined as the average number, of femtocellbase stations and macrocell users per cellsite respectively. A user canexperience outage if the received instantaneous Signal-to-InterferenceRatio (SIR) over a transmission is below a threshold γ. Any feasible(Ñ_(f), Ñ_(c)) can satisfy the outage probability requirements

_(out) ^(f)≦ε,

_(out) ^(c)≦ε in each tier. The outage probabilities

_(out) ^(c)(N_(f), N_(c)). [resp.

_(out) ^(f)(N_(f), N_(c))] can be defined as the probabilities that thedespread narrowband SIR for a macrocell user [femtocell user] at theTier 1 [Tier 2] base station antenna sector is below γ. Assuming the PNcode cross-correlation equals N_(hop)/G (with N_(hop)=G=1, the modelreduces to a non CDMA narrowband transmission; with N_(hop)=G>>1, themodel reduces to a time slotted ALOHA channel), define

$\begin{matrix}{{{{\mathbb{P}}_{out}^{c}\left( {N_{f} \cdot N_{c}} \right)} = {\Pr \left( {{\frac{{G/N_{hop}}P_{r}^{c}}{I_{c,{in}} + I_{c,{out}} + I_{c,f}} \leq \gamma}{{{\hat{\Omega}}_{c}} \geq 1}} \right)}}{{{\mathbb{P}}_{out}^{f}\left( {N_{f} \cdot N_{c}} \right)} = {\Pr \left( {{\frac{{G/N_{hop}}P_{r}^{f}}{{\left( {U - 1} \right) \cdot P_{r}^{f}} + I_{f,f} + I_{f,c}} \leq \gamma}{U \geq 1}} \right)}}} & (3)\end{matrix}$

where |{circumflex over (Ω)}_(c)| can denote the number of points in{circumflex over (Ω)}{circumflex over (Ω_(c))} and the unconditionedU˜Poisson(U_(f)/N_(sec)). The OCs for the macrocell [resp. femtocell]can be obtained by computing the highest N_(f)[N_(c)] for a givenN_(c)[N_(f)], which satisfy a target outage constraint ε. More formally,

Ñ _(f)(N _(c))=sup{N _(f):

_(out) ^(c)(N _(f) ,N _(c))≦ε},Ñ _(c)(N _(f))=sup{N _(c):

_(out) ^(f)(N _(f) ,N _(c))≦ε}.  (4)

The OCs for the two-tier network can be obtained corresponding to thosefeasible combinations of (Ñ_(c), Ñ_(f)) that simultaneously satisfy

_(out) ^(f)≦ε and

_(out) ^(c)≦ε respectively. In doing so, the following theorems can bederived which can quantify the outage probabilities and CCI statisticsin each tier.

Theorem 1: For small femtocell sizes, the statistics of the cross-tierfemtocell CCI I_(c,f) (and intra-tier femtocell CCI I_(f,f)) at a basestation antenna sector can be given by a

${{{Poisson}\mspace{14mu} {SNP}\mspace{14mu} Y} = {\sum\limits_{i \in \varphi_{f}}{Q_{f}\Psi_{i}{X_{i}}^{- \alpha}}}}\mspace{11mu}$ with  iid  $\; {{\Psi_{i} = {\sum\limits_{j = 1}^{U_{i}}\Psi_{ij}}},{10\log_{10}{\left. \Psi_{ij} \right.\sim{\left( {0,\sigma_{dB}^{2}} \right)}}},{{\left. U_{i} \right.\sim U}{U \geq {1\mspace{14mu} {and}\mspace{14mu} {{\left. U \right.\sim{{Poisson}\left( U_{f\;} \right)}}.}}}}}\mspace{14mu}$

In particular, if the outdoor path-loss exponent α=4, then Y follows aLévy-stable distribution with stability exponent ½, whose pdf and cdfcan be given as:

$\begin{matrix}{{{{f_{Y}(y)} = {\sqrt{\frac{\kappa_{f}}{\pi}}y^{{- 3}/2}^{{- \kappa_{f}}/y}}},{{F_{Y}(y)} = {{erfc}\left( \sqrt{\frac{\kappa_{f}}{y}} \right)}}}{where}{\kappa_{f}\overset{\Delta}{=}{\eta_{f}^{2}\pi^{3}{{Q_{f}\left( {E\left\lbrack \Psi^{1/2} \right\rbrack} \right)}^{2}/4.}}}} & (5)\end{matrix}$

In an embodiment of the invention, increasing femtocell size (R_(f)) canincrease the outage probabilities arising from the femtocell CCI I_(f,f)and I_(c,f) in a two-tier network. To elucidate this, observe that anincrease in R_(f) can cause κ_(f) to increase by a factor R_(f) ^(β). Bymonotonicity of erfc(·), the cdf's F_(I) _(f,f) (·), F_(I) _(c,f) (·)can decrease as κ_(f) increases, causing a higher outage probability pertier. Intuitively, a femtocell user located on the edge of a femtocellcan cause excessive CCI at a nearby femtocell base station. The effectof the CCI can appear as a power control penalty factor in equationR_(f) ^(β) equation (5).

Tier 2 users within a femtocell can be assumed to jointly choose ahopping slot. Comparing this against an independent hopping protocol,where users within a femtocell are independently assigned a hoppingslot. With independent hopping, the intensity of Φ_(f) can equal

${\overset{\sim}{\eta}}_{f} = {\frac{\lambda_{f}}{N_{\sec}} \cdot \left( {1 - ^{U_{f}/N_{hop}}} \right)}$

(note the difference of {right arrow over (η)}_(f) from η_(f) inLemma 1) and the average number of interfering users in an activelytransmitting femtocell can equal

$\frac{U_{f}/N_{hop}}{1 - ^{{- U_{f}}/N_{hop}}}.$

With an outage threshold P_(r) ^(f)G/(N_(hop γ)) (equation (3)) at afemtocell base station, two observations are in order:

TH-CDMA transmission: When

${\frac{G}{N_{hop}}1},$

joint hopping can be preferable from an outage probability perspective.Intuitively, joint hopping reduces λ_(f) by a factor N_(hop), which cancause a quadratic decrease in κ_(f) in equation (5); independent hoppingcan decrease the number of interfering users per active femtocell,causing a sub-quadratic decrease in

[Ψ^(1/2)]². The consequence can be that joint hopping can result in agreater decrease in

_(out) ^(f). Using N_(hop)=2, FIG. 7 illustrates this intuition, wherethe gap in outage performance can be dictated by the hotspot userdensity. In FIG. 7, the solid lines 410 represent the joint hoppingperformance when all users within a femtocell share a common hoppingslot. The dotted lines 420 indicate the performance when every femtocelluser is assigned an independent CDMA hopping slot. In heavily loadedfemtocells (U_(f)>>1), a joint hopping scheme can be desirable. Forlightly loaded femtocells

${\eta \simeq {\overset{\sim}{\eta}}_{f} \approx \frac{\lambda_{f}U_{f}}{N_{\sec} \cdot N_{hop}}},$

implying that independent and joint hopping schemes can have nearlyidentical performance.

Random Access transmission: When N_(hop)=G>>1, the femtocell outagethreshold can be P_(r) ^(f)/γ. In this case, it may be preferable to useindependent hopping across the tier 2 network. With joint hopping, evena single interferer within a femtocell can cause outage for the user ofinterest as there may be no interference averaging as illustrated inFIG. 7. An independent hopping scheme can offer increased interferenceavoidance since the likelihood of two femtocell users sharing a hoppingslot can be negligible. Consequently, in non-CDMA two-tier cellularnetworks employing interference avoidance, independent assignment ofhopping slots may be preferable from an outage viewpoint. Using Theorem1, the macrocellular outage probability can be formulated.

Theorem 2 (Macrocell outage probability): Let outdoor path-loss exponentα=4. With Poisson in-cell macrocell CCI I_(c,in), Gaussian out-of-cellCCI I_(c,out) and Lévy-stable femtocell CCI I_(c,f) given by equation(5), the outage probability at the macrocell base station antenna sectorcan be given as:

$\begin{matrix}{{{\varepsilon \geq {\mathbb{P}}_{out}^{c}} = {1 - {\frac{1}{1 - ^{{- \eta_{c}}{\mathcal{H}}}}{\sum\limits_{m = 1}^{\lfloor{\rho_{c}/P_{r}^{c}}\rfloor}{\frac{{^{{- \eta_{c}}{\mathcal{H}}}\left( {\eta_{c}{\mathcal{H}}} \right)}^{m}}{m!}{G_{c}\left( {\overset{\sim}{\rho}}_{c} \right)}}}}}}{where}{{\eta_{c} = \frac{\lambda_{c}}{N_{hop} \cdot N_{\sec}}},\mspace{14mu} {\rho_{c} = \frac{P_{r}^{c}G}{N_{hop} \cdot \gamma}},\mspace{14mu} {{\overset{\sim}{\rho}}_{c} = {\rho_{c} - {\left( {m - 1} \right)P_{r}^{c}}}}}\mspace{14mu} {and}{G_{c}(t)}\overset{\Delta}{=}{\int_{0}^{t}{{f_{I_{c,{out}}}\left( {t - y} \right)}{F_{I_{c,f}}(y)}{{y}.}}}} & (6)\end{matrix}$

Theorems 1 and 2 can provide the tools to quantify the largest N_(f)that can be accommodated at a given N_(c) subject to an outageconstraint ε. The next step is to compute the outage probability at afemtocell as defined in equation (3). To do so, assume that thefemtocell is located on the axis at a distance R₀ from the macrocellcenter and the receive antenna at the femtocell base station is alignedat angle ⊖ with respect to the hexagonal axis (FIG. 6). The followingtheorem can be used to derive a lower bound on the statistics of thetier 1 CCI I_(f,c) at any femtocell located along the hexagonal axis.

Theorem 3 (Lower bound on Macrocellular CCI): For any femtocell basestation, antenna sector located at distance 0<R₀≦R_(c) from themacrocell base station along the hexagonal axis:

-   -   1) The ccdf of the macrocellular interference I_(f,c) over a        femtocell base station antenna sector can be lower bounded as

$\begin{matrix}{{{{{\overset{\_}{F}}_{I_{f,c}}(y)} \geq {1 - {F_{I_{f,c}}^{lb}(y)}}},{{where}\text{:}}}{{F_{I_{f,c}}^{lb}(y)} = {\exp \left\{ {{- \frac{\lambda_{c}}{N_{hop}}}\underset{H_{\sec}}{\int\int}{S\left( {r,\varphi,y} \right)}r{r}{\varphi}} \right\}}}} & (7)\end{matrix}$

where S(r, φ; y)

F _(Ψ)[y/P_(r) ^(c)·(r/|re^(iφ)+R₀|)^(α)], F _(Ψ) is the ccdf of Ψ: 10log₁₀Ψ˜

(0, 2σ_(dB) ²), ⊖ is the femtocell base station antenna alignment angleand

⊂

denotes the region inside the reference macrocell enclosed between⊖≦φ≦⊖+2π/N_(sec).

-   -   2) For a corner femtocell R₀=R_(c) with an omnidirectional        femtocell antenna N_(sec)=1, the ccdf of I_(f,c) can be lower        bounded as

$\begin{matrix}{{{{{\overset{\_}{F}}_{I_{f,c}}(y)} \geq {1 - {F_{I_{f,c}}^{lb}(y)}}},{where}}{F_{I_{f,c}}^{lb}(y)} = {\exp \left\{ {{- 3}\frac{\lambda_{c}}{N_{hop}}\underset{\mathcal{H}}{\int\int}{S\left( {r,{\varphi;y}} \right)}r{r}{\varphi}} \right\}}} & (8)\end{matrix}$

-   -   For a path-loss model, the lower bounds on the femtocell outage        probability can be derived analogously as stated in the        following corollary.

Corollary 1. With the above definitions, assuming a pure path-loss modelwith no shadowing, equations (7) and (8) hold with S(r, φ: y)

1[P_(r) ^(c)·(|re^(iφ)+R₀|/r)^(α)

y]. Theorem 3 can characterize the relationship between the intensity ofmacrocell users and the femtocell outage probability. Observe that theoutage probability F _(I) _(f,c) ^(lb)→1 exponentially, as λ_(c)→∞.Furthermore, increasing N_(hop) can “thin” the intensity of Π_(c),thereby mitigating cross-tier CCI at the femtocell base station. FIG. 8depicts the outage lower bounds to evaluate the impact of macrocellularCCI I_(f,c). Corresponding to an interior and corner femtocell location,the lower bounds can be computed when the femtocell base station antennais either sectored N_(sec)=3 with antenna alignment angle θ=2π/3, orwhen the base station antenna is omnidirectional. No hopping is used(N_(hop)=1), while a unity power ratio (P_(r) ^(f)/P_(r) ^(c)=1) ismaintained. Two observations can be made:

Tightness of lower bound: In one embodiment, the tightness of equations(7) and (8) can show that the cross-tier CCI I_(f,c) can primarily beimpacted by the set of dominant macrocellular interferers. Oneimplication is that one can perform accurate outage analysis at afemtocell by considering only the nearest tier 1 users that individuallycause outage.

Infeasibility of omnidirectional femtocells: In another embodiment, thebenefits of sectored antennas for CCI mitigation at the femtocell basestation can be evident; with a sectored base station antenna, a cornerfemtocell (worst-case macrocell CCI) can perform considerably betterthan an interior omnidirectional femtocell. Using Theorems 1 and 3, thefemtocell outage probability in equation (3) is stated in theorem 4.

Theorem 4 (Femtocell outage probability): Let outdoor path-loss exponentα=4. For small λ_(c), the femtocell outage probability

_(out) ^(f) can be lower bounded as:

$\begin{matrix}{{\varepsilon \geq {\mathbb{P}}_{out}^{f,{l\; b}} \approx {1 - {\frac{^{- U_{f,\sec}}}{1 - ^{- U_{f,\sec}}}{\sum\limits_{m = 1}^{\lfloor{\rho_{f}/P_{r}^{f}}\rfloor}{\frac{U_{f,\sec}^{m}}{m!} \cdot {{G_{f}\left( {\overset{\sim}{\rho}}_{f} \right)}.{Where}}}}}}}{{U_{f,\sec}\overset{\Delta}{=}\frac{U_{f}}{N_{\sec}}},\mspace{14mu} {\rho_{f}\overset{\Delta}{=}\frac{{GP}_{r}^{f}}{N_{hop} \cdot \gamma}},\mspace{14mu} {{\overset{\sim}{\rho}}_{f} = {\rho_{f} - {\left( {m - 1} \right) \cdot P_{r}^{f}}}}}{{{and}\mspace{14mu} {G_{f}(t)}}\overset{\Delta}{=}{{F_{I_{f,f}}(t)} + {\int_{0}^{t}{{f_{I_{f,f}}\left( {t - y} \right)}{\ln \left( {F_{I_{f,c}}^{lb}(y)} \right)}{{y}.}}}}}} & (9)\end{matrix}$

For a given N_(f), Theorem 4 can compute the largest N_(c) which canensure the SIR threshold γ can be satisfied for a fraction (1−ε) of thetime. Furthermore, the lower bound F_(l) _(fx) ^(lb)(·) can be shown tobe tight, hence the computed N_(c) is not overly optimistic. UsingTheorems 2 and 4, the OCs for the two-tier network with interferenceavoidance can now be readily obtained.

IV. FEMTOCELL EXCLUSION REGION AND TIER SELECTION

Suppose the reference macrocell BS has a femtocell exclusion regionR_(f) ^(exc)⊂H surrounding it. This idea may be motivated by the need tosilence neighboring femtocell transmissions which may be strong enoughto individually cause outage at a macrocell base station; similarschemes have been proposed and adopted in the CSMA scheduler in the802.11 standard. The tier 2 femtocell network may then form aheterogeneous SPPP on H with the average number of femtocells in eachcell-site equaling λ_(f)·(|

|−|

_(f) ^(rr)|). The following theorem derives a lower bound on the ccdf ofthe cross-tier femtocell interference I_(c,f) considering the effect ofa femtocell exclusion region.

Lemma 2 (Femtocell exclusion region): With a femtocell exclusion regionof radius R_(f) ^(exc) around the reference macrocell base station, theccdf of cross-tier femtocell CCI I_(c,f) is lower bounded as:

$\begin{matrix}{{{{\overset{\_}{F}}_{I_{c,f}}(y)} \geq {1 - ^{{- {\pi\eta}_{f}}{H{(y)}}}}}{where}{\delta = \frac{2}{\alpha}},\mspace{14mu} {u = {y \cdot \frac{\left( R_{f}^{exc} \right)^{2/\delta}}{Q_{f}}}},\mspace{14mu} {{H( y)}\overset{\Delta}{=} {{\left( \frac{Q_{f}}{y} \right)^{\delta} \begin{pmatrix}{{{E\left\lbrack \Psi^{\delta} \right\rbrack} -}} \\{{F_{\Psi}( u)} {E\left\lbrack {\Psi^{\delta} {\Psi \leq u}} \right\rbrack}}\end{pmatrix}} - {{{\overset{\_}{F}}_{\Psi}( u)} \left( R_{f}^{exc} \right)^{2}}}},{\Psi \overset{\Delta}{=}{\sum\limits_{i = 1}^{U}\Psi_{i}}},\mspace{14mu} {{10\log_{10}{\left. \Psi_{i} \right.\sim{\left( {0,{2\sigma_{dB}^{2}}} \right)}}{and}\; {\left. U \right.\sim X}}{X \geq 1}},{{\left. X \right.\sim{Poisson}}\mspace{11mu} {\left( U_{f} \right).}}} & (10)\end{matrix}$

FIG. 9 depicts the macrocell outage performance as a function of thefemtocell exclusion radius, assuming N_(c)=1, P_(r) ^(f)/P_(r) ^(c)=1.Notice that even a small exclusion radius R_(f) ^(exc) may result in asignificant decrease in

_(out) ^(c). One implication is that a femtocell exclusion region canincrease the number of simultaneous active femtocell transmissions,while satisfying the macrocell outage constraint

_(out) ^(c)≦ε. Once again, the close agreement between analysis andsimulation shows that only the nearby dominant femtocell interferersinfluence outage events at the macrocell base station.

Corollary 2. With no femtocell exclusion (R_(f) ^(exc)=0), the ccdf ofthe cross-tier femtocell CCI I_(c,f) at a macrocell is lower bounded asF _(I) _(c,f) (y)≧1−e^(−πη) ^(f) ^(Q) ^(f) ^(δ) ^(E[Ψ) ^(δ) ^(]y) ^(−δ).

The influence of a femtocell tier selection based handoff policy whereinany tier 1 macrocellular user within the radius R_(f) of a femtocellbase station undergoes handoff to the femtocell will now be considered.In essence, the CCI caused by the nearest macrocell users is mitigated,as these users now employ power control to the femtocell base station.

Lemma 3: With a tier selection policy in which any user within a radiusR_(f) of a femtocell undergoes handoff to the femtocell BS, theintensity of tier 1 users within H after handoff may be given as λ_(c)^(TS)(r)=λ_(c)·e^(−λ) ^(f) ^(πR) ^(f) ² whenever r>R_(f) ^(exc), whereR_(f) ^(exc) is the femtocell exclusion radius.

In one embodiment, for small λ_(f) and r>R_(f) ^(exc), a first orderTaylor approximation may show that λ_(c) ^(TS)≈λ_(c)·(1−λ_(f)πR_(f) ²).One interpretation is that tier-selection offers marginal benefits forsmall femtocell sizes (R_(f)<<R_(c)). Intuitively, a small sizedfemtocell does not cover “enough space” for significant numbers ofmacrocellular users in Ω₂ to accomplish femtocell handoff. However,Theorem 1 shows that a small femtocell size does lead to a lower uplinkoutage probability.

In another embodiment, the network OCs considering the effects of afemtocell exclusion region and tier selection can be obtained byapplying Lemmas 2 and 3 in Theorems 2 and 4 respectively. In doing so,we approximate I_(f,f) as a Poisson SNP whose cdf is described byequation (1).

V. NUMERICAL RESULTS

The following provides the numerical results of an emulation performedin accordance with an embodiment of the present invention. Systemparameters are given in Table I, shown in FIG. 5, and the LabVIEWenvironment was used for numerical simulations. The setup consists ofthe region H surrounded by 18 macrocell sites to consider two rings ofinterferers and 2π/3 sectored antennas at each base station. In equation(10), the statistics of the shadowing gain Ψ were empirically estimatedusing the MATLAB functions ksdensity and ecdf respectively. The OCs wereanalytically obtained using Theorems 1-4 for an outage constraint ε=0.1in equation (4). The following plots compare the OCs for a sharedspectrum network with interference avoidance against a split spectrumnetwork with omnidirectional femtocells.

FIGS. 10 and 11 plot OCs for a macrocell and interior femtocell,respectively, for P^(f) _(r)/P^(c) _(r)=1, 10, 100 and N_(hop)=1. Thefemtocell may use a sectored receive antenna with N_(sec)=3, θ=2π/3. Theclose agreement between the theoretical and empirical OC curvesindicates the accuracy of the analysis. Observe that the outageconstraints oppose one another: increasing P^(f) _(r)/P^(c) _(r)decreases the largest N_(f) sustainable for a given N_(c) from themacrocell BS perspective. From the femtocell standpoint, increasingP^(f) _(r)/P^(c) _(r) increases the largest N_(c) which is sustainablefor a given N_(f).

FIGS. 12 through 14 plot the performance of the shared spectrum networkemploying interference avoidance for a corner and an interior femtocell,as a function of N_(hop) and P^(f) _(r)/P^(c) _(r). In particular, FIG.2 illustrates network OC's for different macrocell-femtocell receivedpower ratios and fixed hopping slots, wherein N_(hop)=4 and N_(sec)=3;and FIGS. 13 and 14 illustrate network OC's with different hoppingslots, wherein P^(f) _(r)/P^(c) _(r)=10 and N_(sec)=3 for a corner andinterior femtocell reference, respectively.

FIG. 12 shows that with P^(f) _(r)/P^(c) _(r)=1 and a lightly loadedtier 1 network, the corner femtocell may achieve greater than 7×improvement in N_(f) relative to the split spectrum network.Intuitively, with P^(f) _(r)/P^(c) _(r)=1, a macrocell base stationtolerates a large cross-tier CCI; one downside being that the femtocellbase station may experience higher macrocellular CCI arising from tier 1users transmitting at maximum power near the cell edge. This may explainwhy N_(f) decreases rapidly with increasing N_(c) in the OC curves for acorner femtocell. With P^(f) _(r)/P^(c) _(r)=10, the OCs for corner andinterior femtocells in FIGS. 13 and 14 offer greater than 2.5×improvement in N_(f) relative to the split spectrum network.Additionally, a greater degree of load balancing can be achieved: withan interior femtocell location, a maximum of N_(c)=45 tier 1 users canbe accommodated. The inference is that in a shared spectrum two-tiernetwork, interference avoidance offers considerable improvement in tier2 femtocell density N_(f) at low N_(c); to achieve load balancing byincreasing N_(c) at the expense of N_(f), an order wise difference inreceive power ratio is required. It may be averred that a practicalwireless system use a larger P^(f) _(r)/P^(c) _(r) closer to the cornerfemtocell relative to the interior; this may ensure that both theinterior and corner femtocells can sustain identical number of tier 1users.

FIG. 15 shows the two-tier OCs when users in each tier employ afemtocell exclusion region and a tier selection policy for femtocellhandoff (P^(f) _(r)/P^(c) _(r)=10, N_(sec)=3, R_(f) ^(rr)=20). One canobserve an increase in N_(f) by up to 10 additional femtocells (or10*U_(f)=50 users) for N_(c)<30 users. Both femtocell exclusion and tierselection do not lead to a higher N_(c). One reason is that a femtocellexclusion region does not alleviate tier 1 CCI at a femtocell.Furthermore, one explanation for the conservative gains in N_(f) may bethat there is a maximum tolerable interference to sustain the outagerequirements at a given femtocell, that may prevent a substantialincrease in the number of actively transmitting femtocells. Next, owingto small femtocell sizes, a tier selection policy may succeed in curbingtier 1 CCI mainly for a large N_(f), which may be sustainable when N_(c)is small (to satisfy

_(out) ^(c)≦ε.). This may explain the dominant gains in N_(f) at alow-to-moderate N_(c). A relevant question is to ask: “How does thesystem capacity with randomly placed users and hotspots compare againsta two-tier network with a given configuration?” Results appear to showthat there is a decline in the system capacity, because theconfiguration contains high levels of cross-tier CCI.

It has been proposed to alleviate cross-tier CCI by varying themacrocell coverage region, through exchanging the pilot channel strengthwith the microcell. Embodiments of the present invention assume thatfemtocells (placed by end consumer) operate with minimal informationexchange with the macrocell base station. Due to reasons of security andscalability—there may be hundreds of embedded femtocells within adensely populated macrocell—handing off unsubscribed users frommacrocell to a femtocell hotspot may not be practical. Moreover,femtocell hotspots have a small radio range (<50 meters). This furthermakes an interference avoidance strategy desirable.

VI. CONCLUSION

The foregoing has presented an uplink capacity analysis and interferenceavoidance strategy for a shared spectrum two-tier DS-CDMA network inaccordance with one embodiment of the present invention. The exactoutage probability at a macrocell base station and tight lower bounds onthe ccdf of the CCI at a femtocell were derived. Interference avoidancethrough a TH-CDMA physical layer coupled with sectorized receiveantennas is shown to consistently outperform a split spectrum two-tiernetwork with omnidirectional femtocell antennas. Considering theworst-case interference at a corner femtocell, the network OCs show a 7×improvement in femtocell density. Load balancing users in each tier maybe achievable through an orderwise difference in receive powers in eachtier. Additional interference avoidance using a femtocell exclusionregion and a tier selection based femtocell handoff may offerconservative improvements in the OCs. The message appears clear:Interference avoidance strategies can make shared spectrum two-tiernetworks a viable proposition in practical wireless systems.

Techniques for Reducing Interference in FDMA Cellular Network

According to embodiments of the present invention, similar techniquesmay be used to reduce interference associated with two-tier networksemploying frequency division multiple access (FDMA) transmission (e.g.,3GPP LTE, 802.16 WiMAX, 3GPP2 UMB, etc.). When FDMA systems use thediscrete Fourier transform and/or its inverse to create and decode thefrequency tones, it is often referred to as orthogonal frequencydivision multiple access (OFDMA) or multiuser OFDM. The term FDMA isgenerally used through this patent with the understanding that OFDMA isa specific type of FDMA and embodiments of the present invention applyto both. According to embodiments of the present invention, one or moreof the following techniques may be provided in order to reduceinterference in FDMA two-tier networks: (1) a spectrum allocation schemeto partition spectrum between cellular and hotspot, or femtocell, basestations; (2) a decentralized spectrum access protocol by femtocells;and (3) a carrier sensing scheme for femtocells to detect idle, orunoccupied, subchannels and avoid creating mutual interference forneighboring femtocells.

According to the first embodiment, a spectrum allocation scheme may beprovided to partition the spectrum between cellular and femtocell basestations. In this embodiment spectrum allocation can maximize thenetwork-wide Area Spectral Efficiency (ASE) in a two-tier network,assuming each macrocell can transmit to a single user per frequencysubchannel, while femtocells may access the spectrum usingFrequency-ALOHA (F-ALOHA), a method to be discussed below. ASE, whichcan be measured in b/s/Hz/m², can be defined as the network-widespatially averaged throughput per frequency subchannel divided by theproduct of the subchannel bandwidth and the area over which thetransmissions take place. The allocation can be determined by qualifyingthe per-tier ASEs based on the propagation environment and neighboringcell interference. With an increasing number of macrocell users, the ASEof the macrocell can either be fixed (for a channel blind scheduler) orincreasing (by opportunistic scheduling). For the femtocell network, theASEs can be derived based on the optimal spectrum access using F-ALOHA.Additionally, the optimal allocation can be determined as one thatmaximizes the weighted mean of the per-tier ASEs. The weights can begiven by the fraction of spectrum accessed by each tier.

In an embodiment of the invention, the expected per-tier throughput canbe derived for the macrocell and femtocell users accounting forinterference from neighboring cells. The maximum ASE of the femtocellnetwork can be shown to be unchanged with addition of hotspots beyond athreshold. At low femtocell densities, a high level of femtocell ASEscan be attained when each femtocell can access most of the availablespectrum. In higher densities, femtocells can use a decreasing fractionof the spectrum; e.g. with an average of 100 femtocells in each cellsite, each femtocell may access 30% of the available spectrum.

In another embodiment of the invention, the proposed spectrum allocationcan allocate the desired level of spatial reuse in a two-tier network,subject to a network-wide QoS requirement, which can create thedesirable expected throughput per-user. Differing QoS constraints canproduce different spectrum allocations due to the competing spatialcoverage scales in each tier. A QoS requiring equal per user throughputsin each tier can mean assigning greater than 90% of spectrum to themacrocell. Conversely, an even division of spectrum can occur when theQoS constraints favor femtocells to provide higher data rates.

Once the spectrum has been allocated, according to one embodiment, adecentralized spectrum access protocol, referred to as Frequency-ALOHA,may be used to access the allocated spectrum. In particular, accordingto one embodiment, if a femtocell transmits over all its allottedsubchannels, it may cause excessive interference to surroundingfemtocells; conversely, accessing only a few subchannels can result in apoor spatial reuse. With Frequency-ALOHA, there can be an optimalfraction of spectrum access for each femtocell in order to maximize thespatial reuse of spectrum, or in effect the net number of simultaneoustransmissions per unit area. The spatial reuse is readily expressibleusing the ASE in b/s/Hz/m².

Alternatively, the spectrum may be allocated based on a carrier sensingscheme, wherein the femtocell may opportunistically detect and transmitover unutilized subchannels, in order to avoid colliding withneighboring femtocells. According to one embodiment, the femtocell maydetect unutilized subchannels by using energy detection in each sub band(or source block in commercial OFDMA standards such as 3GPP LTE).According to one embodiment, the energy detection in a frequencysubchannel may be conducted by measuring the total power (signal plusinterference plus noise) in that subchannel; hypothesis testing can beperformed to decide whether the measured power in the subchannelcorresponds to an occupied subchannel or not. The null hypothesiscorresponds to an unoccupied subchannel which is therefore available fortransmission by the femtocell. The Significant hypothesis corresponds toan occupied subchannel (i.e. there is an ongoing transmission intendedfor either a cellular user or an indoor user in a neighboring femtocellin that subchannel).

The following provides a more detailed description and analysis of usingspectrum allocation and access schemes to reduce interference intwo-tier networks employing FDMA transmission.

I. SYSTEM MODEL

In one embodiment, the cellular system setup can consist of a hexagonalregion H of radius R_(c) with a central macrocell base station Cproviding coverage area

${{\mathcal{H}} = {\frac{3\sqrt{3}}{2}R_{c}^{2}}},$

which can be surrounded by two rings of interfering macrocells. Themacro-cellular network can be overlaid with femtocell hotspots of radiusR_(f), which can be randomly distributed on R² according to ahomogeneous SPPP Ω_(f) with intensity λ_(f). The mean number offemtocells per cell site can be obtained as N_(f)=λ_(f)|H|. Macrocellusers can be assumed to be uniformly distributed inside each cell site.Femtocells can be assumed to provide “closed access” to licensed indoorusers who fall within the radio range R_(f) of their respective homebase stations. Let U=U_(c)+N_(f)U_(f) denote the average number of usersin each cell site. These U users can be distributed into U_(c) uniformlydistributed tier 1 mobile outdoor users and U_(f) users per femtocellhotspot.

A. Per-Tier Spectrum Access

In one embodiment of the invention, the available spectrum can compriseF frequency subchannels each with bandwidth W Hz. To determine theoptimal partitioning (F_(c), F_(f)), where F_(c) subchannels can beavailable for macrocell transmissions and F_(f)=F−F_(c) subchannels maybe available for femtocell transmissions. Denote ρ=F_(c)/F as thefraction of spectrum assigned to the macrocell base station with thefollowing assumptions:

Assumption 1. Each femtocell schedules its users in a round-robin (RR)fashion. The macrocell schedules its users according to either a channelblind RR or a channel aware proportional fair (PF) scheduler.

Assumption 2. The fraction ρ takes a continuum of values in [0, 1] and ρis not quantized for having an integer number of frequency subchannels.Consequently, F_(c)=|ρF| and F_(f)=F−F_(c), where |x| may be used todenote the integer part of a number x.

As illustrated in Assumption 1, the long term expected throughput perindoor user equals 1/U_(f)th of the average femtocell throughput. Thelong term expected macrocell throughput is assumed to be equally dividedamong the U_(c) outdoor users with RR and PF scheduling at themacrocell. With a PF scheduler, this assumption may be reasonableconsidering mobility, which may ensure that all users receive anidentical average Signal to Interference Ratio (SIR) over the long term.

If each femtocell transmits over exactly k frequency subchannels amongtheir allotted F_(f) subchannels, the net portion of accessed spectrumper femtocell equals ρ_(f)(1−ρ) where ρ_(f)

k/F_(f). If femtocells choose their frequency subchannels independentlyand with equal probability, F-ALOHA can effectively “thin” the meannumber of interfering femtocells in each frequency subchannel. Theprobability ρ of a femtocell selecting a given frequency subchannel fortransmission can be given as:

$\begin{matrix}{p = {\frac{\begin{pmatrix}F_{f} \\k\end{pmatrix} - \begin{pmatrix}{F_{f} - 1} \\k\end{pmatrix}}{\begin{pmatrix}F_{f} \\k\end{pmatrix}} = {\frac{k}{F_{f}} = {\rho_{f}.}}}} & (11)\end{matrix}$

Consequently, the set of interfering femtocells per frequency subchannelcan be a marked SPPP Λ_(f) with intensity λ_(f)ρ_(f)=λ_(f)k/F_(f). Whenp_(f)=1, all femtocells in Ω_(f can) access the entire spectrum butmutually interfere in all subchannels. For ρ_(f)<<1, femtocells maytransmit in a small region of spectrum and avoid causing mutualinterference. This strategy can provide a higher spectral efficiencyover each frequency subchannel, but can incur reduced spectrumutilization because femtocells do not transmit over the entire availablespectrum.

B. Channel Model & Variable Rate Transmission

In another embodiment, the downlink channel between each base stationand its users can be .composed of a fixed distance dependent path loss,a slowly varying component modeled by lognormal shadowing and Rayleighfast fading with unit average power. For simplicity, thermal noise maybe neglected at the receiver since cellular systems, by nature, areinterference limited. Although imperfect feedback and/or channelestimation may have a potentially big impact on system capacity, thefollowing does not account for these effects for the sake of analyticaltractability.

Assumption 3. Each user can be assumed to track their SIR in eachsubchannel and feedback the instantaneous rate to their base stationwith zero delay. Further, the channel can support the requested rate asdetermined by the scheduled user with probability 1.

Assumption 4. Base stations assign equal transmission powers to allsubchannels. Each base station may assign rate adaptively based on thereceived SIR per user. Let G denote the Shannon Gap with variable rateM-QAM transmission. Assume an instantaneous transmission rate of b_(i)bps/Hz if the instantaneous SIR lies in |Γ_(l), Γ_(i+1)). Using adaptivemodulation with L discrete rates, the instantaneous rate Wb in a W Hzwide subchannel can be chosen as:

$\begin{matrix}{{b = b_{i}},\mspace{14mu} {{{when}\mspace{14mu} {SIR}}\; \in \left\lbrack {\Gamma_{i},\Gamma_{i + 1}} \right)},{1 \leq i \leq L}} & (12) \\{b_{i} = {{\log_{2}\left( {1 + \frac{\Gamma_{i}}{G}} \right)}\mspace{11mu} {{bps}/{{Hz}.}}}} & (13)\end{matrix}$

Assuming identical statistics over all frequency subchannels, the longterm expected throughput (in b/s/Hz) per macrocell/femtocell in eachsubchannel can be given as:

$\begin{matrix}{T = {{\sum\limits_{l = 1}^{L - 1}{l \cdot {\Pr \left\lbrack {\Gamma_{l} \leq {SIR} < \Gamma_{l + 1}} \right\rbrack}}} + {L \cdot {{\Pr \left\lbrack {{SIR} \geq \Gamma_{L}} \right\rbrack}.}}}} & (14)\end{matrix}$

The expected throughput provided by each macrocell [resp. femtocell] canbe obtained by multiplying the expected throughput in equation (14) bytheir respective spectrum allocation ρ[resp. ρ_(f)(1−ρ)].

II. SPECTRUM ALLOCATION AND PER-TIER EXPECTED THROUGHPUTS

In an embodiment of the invention, let spectrum WF be partitioned suchthat the macrocell base station can transmit over a portion ρ, whilefemtocell base stations may transmit over the remaining 1−ρ fraction ofthe spectrum. Let T_(c)(ρ, U_(c)) be the long term throughput (inb/s/Hz) in each subchannel provided by the macrocell. Obtaining T_(c)can require calculating the average rate per subchannel in equation (14)after spatially averaging the SIR over all locations, and accounting forthe interference from two rings of transmitting macrocells.

Let each femtocell access a portion ρ_(f) of its allotted spectrum usingF-ALOHA, servicing its users in a RR schedule. Define T_(f)(ρ_(f)λ_(f))as the expected femtocell throughput in each frequency subchannel, whichcan be determined by the intensity ρ_(f)λ_(f) of the marked SPPP Λ_(f).With universal frequency reuse across all macrocells, the ASE of themacrocell (resp. femtocell) network can be given as:

$\begin{matrix}{{{ASE}_{c} = \frac{T_{c}\left( {\rho,U_{c}} \right)}{H}},\mspace{14mu} {{ASE}_{c} = {\frac{N_{f}\rho_{f}{T_{f}\left( {\rho_{f}\lambda_{f}} \right)}}{H}.}}} & (15)\end{matrix}$

The factor N_(f)ρ_(f) can represent the mean number of transmittingfemtocells in each subchannel. With bandwidth W, the per-tierthroughputs (in b/s) per subchannel can be calculated by multiplying theASEs in equation (15) by W |H|. The network-wide ASE can therefore begiven as:

$\begin{matrix}{{ASE} = {{{\rho \; {ASE}_{c}} + {\left( {1 - \rho} \right){ASE}_{f}}}\mspace{45mu} = {{\frac{1}{H}\rho \; {T_{c}\left( {\rho,U_{c}} \right)}} + {\left( {1 - \rho} \right)N_{f}\rho_{f}{{T_{f}\left( {\rho_{f}\lambda_{f}} \right)}.}}}}} & (16)\end{matrix}$

The expected network throughput (in b/s) over the WF wide spectrum canbe obtained by multiplying equation (16) by WF |H|. Before determiningthe spectrum allocation, QoS may first be stipulated to require η, whichcan ensure that users in either tier are guaranteed a minimum expectedthroughput. By implication, η also can regulate the maximum amount ofspectrum that any tier can receive.

Definition 1. The QoS parameter η can guarantee that the expectedthroughput per user in one tier is at least η/(1−η) with respect to theother tier. Choosing a different η can enable assigning differentpriorities (QoS) to one tier relative to the other. For example, settingη=0.5 can ensure that users in both tiers obtain identical expectedrates. On the other hand, decreasing η can favor assigning greaterspectrum to the tier providing a higher expected throughput per activeuser.

Given a total available spectrum of 1 Hz, the problem can be todetermine the optimal spectrum allocation η over all possible spectrumpartitioning strategies ωε[0, 1] between the macrocell and femtocells.In an embodiment, the spectrum allocation can maximize the network-wideASE with a QoS constraint η on the minimum expected per-tierthroughput/user, as shown below:

$\begin{matrix}\left. {{\rho = {{\frac{1}{H}\arg {\max\limits_{0 \leq \omega \leq 1}{\omega \; {T_{c}\left( {\omega,U_{c}} \right)}}}} + {\left( {1 - \omega} \right)N_{f}\rho_{f}{T_{f}\left( {\rho_{f}\lambda_{f}} \right)}}}}{{{subject}\mspace{14mu} {to}\mspace{14mu} \min \left\{ {{T_{c,u}(\omega)},{T_{f,u}(\omega)}} \right\}} \geq {{\eta \; {T_{c,u}(\omega)}} + {T_{f,u}(\omega)}}}} \right) & (17) \\{{{{where}\mspace{14mu} {T_{c,u}(\omega)}}\overset{\Delta}{=}\frac{\omega \; {T_{c}\left( {\omega,U_{c}} \right)}}{U_{c}}}\mspace{14mu} {{{and}\mspace{14mu} {T_{f,u}(\omega)}}\overset{\Delta}{=}{\frac{\left( {1 - \omega} \right)\rho_{f}{T_{f}\left( {\rho_{f}\lambda_{f}} \right)}}{U_{f}}.}}} & (18)\end{matrix}$

Here T_(c,u)(ω) and T_(f,u)(ω) are the expected throughputs for amacrocell and femtocell user respectively. Whenever the averagesubchannel throughput T_(c)(ω, U_(c)) is independent of the spectrum ωassigned to the macrocell, the objective function in equation (17) canbe an affine function with respect to ω. The following propositionderives the optimizing ρ considering the maximum may be obtained at theextremal points of the constraint set:

In an embodiment, if the expected macrocell throughput per subchannel isindependent of the total spectrum allocated to the macrocell ω, i.e.T_(c)(ω, U_(c))=T_(c)(U_(c))^(∀)ωε|0, 1|, the optimizing ρ in equation(17) can satisfy the QoS constraint with equality, belonging to a setwith two candidate spectrum allocation assignments can be given as:

$\begin{matrix}{{\rho^{*} \in \left\{ {x,{1 - x}} \right\}},{x\overset{\Delta}{=}{\left\lbrack {1 + {\frac{1 - \eta}{\eta}\frac{T_{c}\left( U_{c} \right)}{U_{c}}\frac{U_{f}}{\rho_{f}{T_{f}\left( {\rho_{f}\lambda_{f}} \right)}}}} \right\rbrack^{- 1}.}}} & (19)\end{matrix}$

Since T_(c)(ω, U_(c))=T_(c)(U_(c))^(∀)ωε|0, 1|, the optimization problemin equation (17) may be to determine the optimal ρ which can maximize aconvex combination of T_(c) and N_(f) ρ_(f)T_(f)(ρ_(f)λ_(f)) with alinear constraint. Consequently, the argument maximizer can be locatedat the external points of the constraint set in equation (18). Solvingfor the ρ which satisfies equation (18) with equality can yield equation(19).

Without a QoS requirement (allowing η→0), the objective function inequation (17) can be a convex combination of the macrocell and femtocellthroughputs which can be maximized at the extreme points ρε{0, 1}. Sucha partitioning can be unfair since it can result in a greedy allocationof the entire spectrum to one tier. For a generic macrocellscheduler—when the foregoing proposition maybe applicable—equation (17)is one dimensional optimization problem that can be solved efficientlyfor a given η using a numerical search.

A. Macrocell Throughput: RR Scheduling.

Assuming that the central macrocell base station C₀ in the hexagonalregion H is placed at the origin, the normalized positions of theinterfering base stations C_(k)εβ, k=1 . . . 18 are represented in polarform b_(k), k:εβ using MATLAB notation as:

$\begin{matrix}{{b_{k}/R_{c}} \in {\left\{ {\sqrt{3}^{{({{\pi/6} + {{\lbrack{0:5}\rbrack}{\pi/3}}})}}} \right\}\bigcup\left\{ {3^{{({{\lbrack{0:5}\rbrack}{\pi/3}})}}} \right\}\bigcup\left\{ {2\sqrt{3}^{{({{\pi/6} + {{\lbrack{0:5}\rbrack}{\pi/3}}})}}} \right\}}} & (20)\end{matrix}$

Let h₀ denote the Rayleigh fast fade (with exponentially distributedchannel power |h₀|² with unit mean) between the central macrocell basestation C₀ and its scheduled user 0. Denoting the Euclidean norm by ∥·∥,the expression for the received SIR for macrocell user 0 at position rmay be given as:

$\begin{matrix}{{{{SIR}_{c}(r)} = {\frac{\Theta_{0}}{\Psi_{I}(r)}{h_{0}}^{2}{{r/R_{c}}}^{{- \alpha_{c}}\;}}}{where}} & (21) \\{{\Psi_{I}(r)} = {\sum\limits_{k \in \mathcal{B}}{{\frac{r - b_{k}}{R_{c}}}^{- \alpha_{c}}{h_{0k}}^{2}\Theta_{0k}}}} & (22)\end{matrix}$

Here α_(c) represents the outdoor path loss exponent and|h_(0k)|²˜exp(1) is the exponentially distributed channel power betweeninterfering BS C_(k) and the user of interest. The RV Θ₀ [resp. Θ_(0k)]is the lognormal shadowing between the central BS [resp. interferingBSs] and the desired user, which are distributed asLN(ζμ_(c,dB)·ζ²σ_(c,dB) ²), where ζ=0.1 ln 10 is a scaling constant. Foranalytical tractability the following assumption may be made regardingthe distribution of a composite lognormal-exponential RV:

Assumption 5: The distribution of a composite lognormal-exponential RVΘ_(k)|h_(0k)|² is modeled as a lognormal distribution using Turkmani'sapproximation:

$\begin{matrix}{{f_{\Psi_{I}}(x)} = {\frac{1}{x\; \sigma_{i}\sqrt{2\pi}}{\exp\left\lbrack \frac{- \left( {{\ln \; x} - \mu_{i}} \right)^{2}}{2\sigma_{i}^{2}} \right\rbrack}}} & (23) \\{{\mu_{i} = {\zeta \left( {\mu_{c,{dB}} - 2.5} \right)}},\mspace{14mu} {\sigma_{i} = {\zeta \sqrt{\sigma_{c,{dB}}^{2} + 5.57^{2}}}}} & (24)\end{matrix}$

Assumption 6: For a fixed r, using the moment generating function basedtechnique described in N. B. Mehta J. Wu, A. F. Molisch, and J. Zhang,“Approximating a sum of random variables with a lognormal,” IEEE Trans.On Wireless Communication., vol. 6, no. 7, pp. 2690-2699, July 2007(Mehta et al.), the contents of which are hereby incorporated herein byreference in their entirety, the sum of |β| independent, but notidentically distributed lognormal RVs in the expressionΨ_(l)(r)=Σ_(kCβ)(∥r−b_(k)∥/R_(c))^(−σ) ^(c) Ψ_(k) can be approximated bya single lognormal RV with parameters LN(μ_(I)(r); σ_(I) ²(r)). UsingAssumptions 5-6 and equation (21), SIR_(c)(r) may be distributedaccording to a lognormal RV LN(μ_(C)(r), σ_(C) ²(r) whereμ_(C)(r)=μ_(S)(r)−μ_(l)(r) and σ_(C)(r)=√{square root over (σ_(S)²(r)+σ_(l) ²(r))}{square root over (σ_(S) ²(r)+σ_(l) ²(r))}. Then, thedistribution of the SIR for a mobile tier 1 user at position r withrespect to the central BS may be given by

$\begin{matrix}{{{\Pr \left\lbrack {{{{SIR}_{c}(r)} \leq \Gamma}r} \right\rbrack} = {1 - {Q\left\lbrack \frac{{\ln \left( {\Gamma {{r/R_{c}}}^{\alpha_{c}}} \right)} - \mu_{C}}{\sigma_{C}} \right\rbrack}}}{where}{{Q(x)}\overset{\Delta}{=}{\frac{1}{\sqrt{2\pi}}{\int_{x}^{\infty}{^{{- t^{2}}/2}{t}}}}}} & (25)\end{matrix}$

is the complementary cumulative distribution function (CCDF) of astandard normal. Defining

${a(r)}\overset{\Delta}{=}\frac{{\ln \; \Gamma} - {\mu_{C}(r)}}{\sigma_{C}(r)}$and ${b\overset{\Delta}{=}\frac{\alpha_{C}}{\sigma_{C}(r)}},$

equation (25) simplifies to

Pr[SIR _(c)(r)≦Γ|r]=1−Q[a(r)+b(r)ln∥r/R _(c)∥]  (26)

Averaging equation (26) over a hexagonal cell region may be difficult.Alternatively, the spatially averaged CDF of SIR_(c) can be obtainedapproximately by considering a circular region of radius

$\sqrt{\frac{}{\pi}},$

which results in the same area as the cell site H. To calculate thespatial throughput inside this circular region, the region may bedivided into M non-overlapping annuli. For tractability, a simplifyingassumption is that all users inside an annulus experience identicalshadowing statistics (i.e. identical μ_(C)(r) and σ_(C)(r)). Denotingthe distance of the user from C₀ by ∥r∥=R, the following lemma derivesthe expected spatial throughput by averaging SIR_(c)(R) inside acircular annulus with inner radius R₁ and outer radius R₂.

Lemma 1: The spatially averaged SIR distribution inside a circularannulus with inner radius R₁ and outer radius R₂ may be given as:

$\begin{matrix}{{{_{R}\left\lbrack {\Pr \begin{pmatrix}{{{SIR}_{c} \leq \Gamma}} \\{R_{1} \leq R \leq R_{2}}\end{pmatrix}} \right\rbrack} = {1 - {\frac{1}{\left( {R_{2}^{2} - R_{1}^{2}} \right)}\begin{bmatrix}{{R_{2}^{2}{C\left( {a_{2},b} \right)}} -} \\{R_{1}^{2}{C\left( {a_{1},b} \right)}}\end{bmatrix}}}}{where}} & (27) \\{{C\left( {a,b} \right)}\overset{\Delta}{=}{{Q(a)} + {{\exp\left( \frac{2 - {2{ab}}}{b^{2}} \right)}{Q\left( \frac{2 - {ab}}{b} \right)}}}} & (28) \\{{a\overset{\Delta}{=}\frac{{\ln \; \Gamma} - {\mu_{C}\left( R_{2} \right)}}{\sigma_{C}\left( R_{2} \right)}},{b\overset{\Delta}{=}\frac{a_{c}}{\sigma_{C}\left( R_{2} \right)}}} & (29) \\{{a_{2} = {a + {b\; {\ln \left( {R_{2}/R_{c}} \right)}}}},{a_{1} = {a + {b\; {\ln \left( {R_{1}/R_{c}} \right)}}}},} & (30)\end{matrix}$

Lemma 1 provides a simple method for estimating the cell-averagedmacrocell throughput per sub-channel. The probability that a user liesin an annulus with inner radius R_(m−1) and outer radius R_(m) (1≦m≦Mwith R₀=0) equals

$\frac{\pi \left( {R_{m}^{2} - R_{m - 1}^{2}} \right)}{}.$

Assumptions 5 through 6 may be used for computing the shadowingparameters σ_(C) and μ_(C) at discrete locations R_(m,l)≦m≦M where

$R_{M} = {\sqrt{\frac{}{\pi}}.}$

The spatially averaged SIR distribution for a macrocell user maytherefore, be approximated as follows:

$\begin{matrix}{{\Pr \left( {{SIR}_{c} \leq \Gamma} \right)} = {_{R}\left\lbrack {\Pr \left( {{{SIR}_{c}(R)} \leq \Gamma} \right)} \right\rbrack}} & (31) \\{\mspace{140mu} {= {1 - {\frac{\pi \; R_{1}^{2}}{}{C\left( {{a_{1} + {b_{1}\ln \; \frac{R_{1}}{R_{c}}}},b_{1}} \right)}}}}} & (32) \\{\mspace{149mu} {{- {\sum\limits_{m = 2}^{M}{\frac{\pi}{}\begin{bmatrix}{{R_{m}^{2}{C\left( {{a_{m} + {b_{m}\ln \; \frac{R_{m}}{R_{c}}}},b_{m}} \right)}} -} \\{R_{m - 1}^{2}{C\left( {{a_{m} + {b_{m}\ln \frac{R_{m - 1}}{R_{c}}}},b_{m + 1}} \right)}}\end{bmatrix}}}}{where}{{a_{m}\overset{\Delta}{=}{{\frac{{\ln \; \Gamma} - {\mu_{C}\left( R_{m} \right)}}{\sigma_{C}\left( R_{m} \right)}\mspace{14mu} {and}\mspace{14mu} b_{m}}\overset{\Delta}{=}{a_{c}/{\sigma_{C}\left( R_{m} \right)}}}},}}} & (33)\end{matrix}$

where (32) approximates (31) by spatially averaging SIR_(c) over Mdifferent annulus. Equation (33) is obtained by substituting (27) insidethe conditional expectation in (32) and the corresponding probabilitythat the user lies in annulus m, 1≦m≦M. Combining equations (14) and(31), the average macrocell throughput T_(c) in a given subchannel isexpressed as

$\begin{matrix}{T_{c} = {{{\sum\limits_{l = 1}^{L - 1}{l \cdot {_{R}\left\lbrack {\Pr \left( {\Gamma_{l} \leq {{SIR}_{c}(R)} < \Gamma_{l + 1}} \right)} \right\rbrack}}} + \mspace{56mu} {L \cdot {_{R}\left\lbrack {\Pr \left( {{{SIR}_{c}(R)} \geq \Gamma_{L}} \right)} \right\rbrack}}}\mspace{25mu} = {{\sum\limits_{l = 1}^{L - 1}{l \cdot \begin{pmatrix}{{_{R}\left\lbrack {\Pr\left( {{{SIR}_{c}(R)} \leq \Gamma_{l + 1}} \right)} \right\rbrack} -} \\{ {_{R}\left\lbrack {\Pr\left( {{{SIR}_{c}(R)} < \Gamma_{l}} \right)} \right\rbrack}}\end{pmatrix}}} + {L \cdot {_{R}\left\lbrack {\Pr\left( {{{SIR}_{c}( R)} > \Gamma_{L}} \right)} \right\rbrack}}}}} & (34)\end{matrix}$

FIG. 16( a) plots T_(c) (in b/s/Hz) with RR scheduling as a function ofthe outdoor path-loss exponent α_(c) for the system parameters in FIG.23. The close agreement between theory and numerical simulationsindicates that the theoretically obtained SIR distribution is anaccurate approximation for practical throughput in a macrocellularenvironment.

B. Macrocell Throughput: PF Scheduling.

In another embodiment of the invention, in contrast to a RR scheduler, aPF scheduler can enable macrocell users to compete for resources basedon their requested rates normalized by their average throughput.Consequently, the macrocell can select the user with the highest raterelative to their average rate. During the transmission interval n insubchannel m, denote R_(k)|m, n| as the requested rate for user k,1≦k≦U_(c), located at position r_(k) with respect to the centralmacrocell C. Let R _(k)[n] windowed mean throughput obtained by user kover the F_(c) frequency subchannels allocated for macrocelltransmission. The PF scheduler can select the k whose currentsupportable rate may be high relative to their mean rate. The schedulingpolicy per subchannel m with equal per-subchannel transmission powers(Assumption 4) can be described as:

$\begin{matrix}{{\overset{\sim}{k}\left( {m,n} \right)} = {\arg \; {\max\limits_{1 \leq k \leq U_{c}}{\frac{R_{k}{{m,n}}}{{\overset{\_}{R}}_{k}{n}}.}}}} & (35)\end{matrix}$

Note that mobile user k calculates R_(k) |m, n| using equations (12) and(21), respectively. The windowed throughput per user prior totransmission interval (n+1) can be updated according to the followingrule,

$\begin{matrix}{{{{\overset{\_}{R}}_{k}{{n + 1}}} = {{\left( {1 - \frac{1}{N}} \right){\overset{\_}{R}}_{k}{n}} + {\frac{1}{N}{\sum\limits_{m = 1}^{F_{c}}{R_{k}{{m,n}}1{{k\mspace{95mu} = {\overset{\sim}{k}\left( {m,n} \right)}}}}}}}},{1 \leq k \leq U_{c}},} & (36)\end{matrix}$

where 1|·| is the indicator function determining whether user k isscheduled during transmission interval n in frequency link or not. Thewindow size N can be a parameter that can be selected considering thedelay tolerance for each user. Choosing a smaller N can enable a givenuser to be scheduled more often, whereas choosing larger N can relax thefairness constraint and can allow the scheduler to wait longer beforescheduling a user. By the strong law of large numbers, the averagethroughput per frequency subchannel for a given set of user positionscan be obtained from the sample average over a long duration andexpressed as:

$\begin{matrix}{{E {{\overset{\_}{R}\left( {F_{C}, U_{C}} \right)} {{r_{1}, {\ldots \mspace{14mu} r_{U_{C}}}}} {\lim\limits_{n\rightarrow\infty}{\frac{1}{n} {\sum\limits_{j = 1}^{n}{\sum\limits_{m = 1}^{F_{C}}\frac{R_{\overset{\sim}{k}}{{m,j}}}{F_{C}}}}}}}}, {\overset{\sim}{k} \in \left\{ {1, 2, {\ldots \mspace{14mu} U_{C}}} \right\}},} & (37)\end{matrix}$

where the expectation on the left hand side can be over the joint pdf ofall channel gains between users and their serving and interfering basestations. The spatial averaged subchannel macrocell throughput can beobtained by averaging equation (37) with respect to the joint pdf f_(R)₁ , . . . R_(U) _(c) (·) and can be given as:

$\begin{matrix}{{{T_{C}\left( {\rho,U_{C}} \right)} = {{E_{R_{1},{\ldots R}_{U_{c}}}{E}{\overset{\_}{R}\left( {F_{C},U_{C}} \right)}R_{1}} = r_{1}}},{{\ldots \mspace{14mu} R_{U_{c}}} = \left. r_{U_{c}}||. \right.}} & (38)\end{matrix}$

Using equation (38) to compute T_(c)(ρ, U_(c)) can be analyticallyintractable. In an embodiment of the invention, numerical simulation canbe used to empirically estimate Tc(ρ, U_(c)), which can be used toderive the bandwidth partitioning. In the simulation, the number ofsubchannels can be set as F_(c)=1 with a link bandwidth W=15 KHz and aPF window parameter N=500 OFDM symbols. Each mobile can be moving atν=13.34 m/s (30 mph) and the per-link throughput of equation (37) can beaveraged over 500 drops, with 8000 trials/drop for modeling time-varyingRayleigh fading—which can model propagation environments with scatteringand no direct line-of-sight component. The Rayleigh fading can be heldfixed over a duration Tc=0.4/f_(d) where

$f_{d} = \frac{{vf}_{c}}{3*10^{8}}$

is the Doppler frequency at a carrier frequency f_(c)=2 GHz. FIG. 16( b)compares the performance of PF (numerically evaluated) versus RRscheduling for different U_(c) (with α_(c)=4). Exploiting channelvariations through proportional fairness can roughly double the expectedsubchannel throughput.

C. Femtocell Throughput

In another embodiment, since femtocells can be modeled as randomlyplaced on R² according to a SPPP Φ_(f) with intensity λ_(f), theinterference experienced by a femtocell user can depend on the distancesof these interfering base stations relative to the user and theirrespective channel gains. Using F-ALOHA, the interfering femtocells canform a marked SPPP Λ_(f) ⊂Φ_(f) with intensity ρ_(f)λ_(f). In a givenfrequency subchannel, the cochannel interference I_(f,f) experienced bya user 0 within femtocell F₀ can be given as:

$\begin{matrix}{{I_{f,f} = {\sum\limits_{k \in \Lambda_{f}}{A_{f}\Theta_{0k}{h_{0k}}^{2}{x_{0k}}^{- \alpha_{f}}}}},} & (39)\end{matrix}$

where Θ_(0k)˜LN(ζμ_(fo,dB:)ζ²σ_(fo,dB) ²) represents the lognormalshadowing from femtocell F_(k) to user 0 and |h_(0k)|² is theexponentially distributed channel power between interfering femtocellF_(k) and user 0 inside F₀. Denoting the exponentially distributedchannel power between F₀ and user 0 as |h₀|², the received SIR can begiven as:

$\begin{matrix}{{SIR}_{f} = {\frac{B_{f}\Theta_{0}{h_{0}}^{2}{R_{f}}^{- \beta_{f}}}{\sum\limits_{k \in \Lambda_{f}}{A_{f}\Theta_{0k}{h_{0k}}^{2}{x_{0k}}^{- \alpha_{f}}}}.}} & (40)\end{matrix}$

Here, user 0 is assumed to be on the edge of the home femtocell F₀ andx_(0k) represents the locations of the interfering femtocells F_(k) withrespect to user 0. The term Θ₀˜LN(ζμ_(fi,dB), ζ⁻²σ_(fi,dB) ²) is theindoor lognormal shadowing, and Ψ₀

Θ₀|h₀|² [resp. Ψ_(0k)

Θ_(0k)|h_(0k)|²] is the effective channel gains from the desired [resp.interfering base station]. The terms α_(f) [resp. β_(f)] represent thepath-loss exponents resulting from interfering transmissions [resp.in-home transmissions] to the user of interest. A simple model can beused to distinguish between the fixed losses arising from in-home andinterfering transmissions. Specifically, home users can be insulatedagainst interfering femtocell transmissions through double penetrationlosses arising from external wall partitions. Consequently, A_(f) andB_(f) (in dB) are related as A_(f,dB)=B_(f,dB)+2P_(f,dB) where

$P_{f} = \sqrt{\frac{A_{f}}{B_{f}}}$

is the wall penetration loss.

Using Assumption 5, The channel gain Θ₀|h₀|² can be approximated as alognormal rv Ψ₀˜LN(μ_(S), σ_(s) ²). Similarly, the channel gainsΘ_(0k)|h_(0k)|²□k can be approximated as iid rv's distributed asΨ₀˜LN(μ_(l), σ_(l) ²). Equation (40) then can be simplified to:

$\begin{matrix}{{SIR}_{f} = {\frac{\Psi_{0}{R_{f}}^{- \beta_{f}}}{\sum\limits_{k \in \Lambda_{f}}{P_{f}^{2}\Psi_{0k}{x_{k}}^{- \alpha_{f}}}}.}} & (41)\end{matrix}$

The closed form distribution of the Poisson SNP

$I_{f,f} = {\sum\limits_{k \in \Lambda_{f}}{P_{f}^{2}\Psi_{0k}{x_{k}}^{- \alpha_{f}}}}$

is known when α_(f)=4. However, tight lower bounds on Pr(I_(f,f)>y) canbe obtained by considering femtocells whose interference individuallyexceeds y. Using this idea, the following lemma can provide anasymptotically tight lower bound on the tail distribution of I_(f,f).

Lemma 2. With randomized transmissions and lacking power control, thelower bound on distribution I_(f,f) can be given as:

Pr(I _(f,f) >y)≧1−exp|−πλ_(f)ρ_(f) E|Ψ _(l) ^(δ) ^(f) |P _(f) ^(2δ) ^(f)y ^(−δ) ^(f) |,  (42)

where

$\delta_{f}\overset{\Delta}{=}{\frac{2}{\alpha_{f}}.}$

When α_(f)=4, I_(f,f) can be distributed as:

$\begin{matrix}{{\Pr \left( {I_{f,f} > y} \right)} = {1 - {{{erfc}\left( \frac{\pi^{3/2}\lambda_{f}\rho_{f}P_{f}E{\Psi^{1/2}}}{2\sqrt{y}} \right)}.}}} & (43)\end{matrix}$

Lemma 2 can provide the relationship between the density λ_(f)ρ_(f) ofinterfering femtocells in Λ_(f) and the distribution of the CCI at afemtocell. For fixed y, as ρ_(f)→0, the tail probability Pr(I_(f,f)>y)→0in equation (42) indicating that selecting fewer subchannels usingF-ALOHA transmission can provide greater resilience against persistentcollisions from nearby femtocells. The distribution of the femtocell SIRin equation (40) can be obtained as:

$\begin{matrix}{{\Pr \left( {{SIR}_{f} \leq \Gamma} \right)} = {\Pr\left( {\frac{\Psi_{0}{R_{f}}^{- \beta_{f}}}{\sum\limits_{k \in \Lambda_{f}}{P_{f}^{2}\Psi_{0k}{x_{k}}^{- \alpha_{f}}}} \leq \Gamma} \right)}} & (44) \\{\mspace{135mu} {= {E_{\Psi_{0}}\left\lbrack {\Pr \begin{pmatrix}{{\sum\limits_{k \in \Lambda_{f}}{P_{f}^{2}\Psi_{0k}{x_{k}}^{- \alpha_{f}}}} \geq} \\{{\frac{\psi_{0}{R_{f}}^{- \beta_{f}}}{\Gamma}\Psi_{0}} = \psi_{0}}\end{pmatrix}} \right\rbrack}}} & (45) \\{\mspace{135mu} {\geq {1 - {E_{\Psi_{0}}\left\{ {\exp\left\lbrack {{- {\pi\lambda}_{f}}\rho_{f}E{\Psi_{1}^{\delta_{f}}}\left( \frac{P_{f}^{2}\Gamma}{\psi_{0}{R_{f}}^{- \beta_{f}}} \right)^{\delta_{f}}} \right\rbrack} \right\}}}}} & (46) \\{\mspace{135mu} {= {1 - {E_{\Psi_{0}}{{\exp\left( {{- \rho_{f}}k_{f}\Gamma^{\delta_{f}}\Psi_{0}^{- \delta_{f}}} \right)}}}}}} & (47)\end{matrix}$

where, κ_(f)

π{circumflex over (λ)}_(f)E|Ψ_(f) ^(δ) ^(f) |(P_(f) ²|R_(f)|^(ρ) ^(f))^(δ) ^(f) , where equations (45) and (46) follow by conditioning on Ψ₀,assuming independence of Ψ₀ and Ψ_(0k) ∀k:εΛ_(f), and applying equation(42). Although it may not be possible to obtain a closed form expressionfor the expectation in equation (47), the distribution SIR_(f) can becalculated numerically. The mean subchannel throughput T_(f) can becalculated by combining equations (14) and (47):

$\begin{matrix}{{T_{f}\left( {\rho_{f}\lambda_{f}} \right)} = {{\sum\limits_{i = 1}^{L - 1}{l \cdot {\Pr \left( {\Gamma_{i} \leq {SIR}_{f} < \Gamma_{i + 1}} \right)}}} + {L \cdot {\Pr \left( {{SIR}_{f} \geq \Gamma_{i + 1}} \right)}}}} & (48) \\{\approx {{\sum\limits_{i = 1}^{L - 1}{{l \cdot E_{\Psi_{0}}} {\begin{matrix}{{\exp\left( {{- \rho_{f}}\kappa_{f}\Gamma_{i + 1}^{\delta_{f}}\Psi_{0}^{- \delta_{f}}} \right)} -} \\{\exp\left( {\rho_{f}\kappa_{f}\Gamma_{l}^{\delta_{f}}\Psi_{0}^{- \delta_{f}}} \right)}\end{matrix}}}} + {{L \cdot E_{\Psi_{0}}} {{{\exp\left( {{- \rho_{f}} \kappa_{f} \Gamma_{L}^{\delta_{f}} \Psi_{0}^{- \delta_{f}}} \right)} }.}}}} & (49)\end{matrix}$

The approximation in equation (48) is because the right-hand side inequation (47) is a lower bound on Pr(SIR_(f)≦Γ). FIG. 17 plots thefemtocell throughput (1−ρ)ρ_(f)T_(f) (in b/s/Hz) assuming the entirebandwidth is allocated to femtocells (ρ=0). Two cases are considered,namely (1) high attenuation (marked “HA” with α_(f)=4, P_(f,dB)=10) and(2) low attenuation (marked “LA” with α_(f)=3.5, P_(f,dB)=2) fromneighboring femtocells. Setting ρ_(f)=1 and assuming N_(f)=50femtocells/cell site, the femtocell throughput falls from approximately4.5 b/s/Hz in a HA environment to nearly 0.5 b/s/Hz in an LA scenario,including the sensitivity of femtocell throughput to propagation fromnearby femtocells.

To calculate the optimum ρ_(f), the maximum ASE per subchannel may bemaximized. This can be analogous to answering the question: Whatfraction of subchannels should each femtocell access to maximize spatialreuse? At this ρ_(f), the F-ALOHA access by each femtocell can beoptimally traded off against neighboring femtocell interference in eachsubchannel. Mathematically, ρ_(f) is the solution to the followingoptimization problem:

$\begin{matrix}{\rho_{f}^{*} = {\lambda_{f}\arg \; {\max\limits_{0 \leq \theta \leq 1}{\theta \; {T_{f}\left( {\theta\lambda}_{f} \right)}}}}} & (50) \\{{ASE}_{f}^{*} = {\rho_{f}^{*}\lambda_{f}{{T_{f}\left( {\rho_{f}^{*}\lambda_{f}} \right)}.}}} & (51)\end{matrix}$

To justify equation (50), observe that there can be an average of|H|ρ_(f)λ_(f) transmitting femtocells per subchannel. With F-ALOHAaccess of 0<⊖≦1, each femtocell can obtain an average subchannelthroughput of T_(f)(⊖), which can result in ASE_(f) equalingλ_(f)⊖·T_(f)(⊖λ_(f)). Alternatively, given any allocation ρ, equation(50) can compute the F-ALOHA spectrum access ρ_(f) which can maximizethe mean overall throughput (1−ρ)ρ_(f)T_(f)(ρ_(f)λ_(f)) per femtocell.

In one embodiment, the ASE in equation (50) only depends on theeffective intensity λ_(f)⊖ of interfering femtocells per subchannel.With increasing λ_(f), provided ρ_(f)*<1, the intensity of Λ_(f) can begiven as λ_(f)ρ_(f)* and remains constant, implying that the optimalρ_(f) is a monotone decreasing function of λ_(f). Consequently, ifρ_(f)<1 for a given λ_(f), the maximum ASE per subchannel is fixed. Thismeans that with increasing λ_(f), the network-wide femtocell throughputequaling

W F·(1−ρ)ASE_(f)* grow linearly with (1−ρ).

FIG. 18 a illustrates a plot of equation (50) for different N_(f) withα_(f)=3.5 and P_(f,dB)=2. In all cases, the highest ASE is fixed atnearly 0.000121 b/s/Hz/m², indicating that the spatial reuse provided byfemtocells does not increase further when the number of femtocellsincreases from an average of 50 femtocells/cell site to 100femtocells/cell site. With a low femtocell density (N_(f)=10), adesirable strategy can be to access the entire spectrum from bandwidthpartitioning. In a dense network (N_(f)=100), the ASE can be maximizedwhen each femtocell accesses approximately 30% of the availablespectrum. Furthermore, in equation (50), as long as ρ_(f)*=1, eachfemtocell can access the entire available spectrum (1−ρ), consequentlyT_(f) decreases with addition of femtocells. However, if ρ_(f)*<1,T_(f)=ASE_(f)/(λ_(f)ρ_(f)*) remains constant with increasing λ_(f) (seeFIG. 18( b)). However, as λ_(f)→∞, since ρε(0, 1), the mean overallthroughput per femtocell approaches zero, as the following limit shows:

$\begin{matrix}{{{\lim\limits_{\lambda_{f}\rightarrow\infty}{{T_{f}\left( {1 - \rho} \right)}\rho_{f}}} \leq {\lim\limits_{\lambda_{f}\rightarrow\infty}{T_{f}\rho_{f}}}} = 0.} & (52)\end{matrix}$

One may explore the dependence of the mean overall femtocell throughputT_(f)ρ_(f) (1−ρ) on the spectrum allocation ρ and F-ALOHA access ρ_(f).Equivalently: With increasing femtocell density λ_(f), can increasingallocated spectrum (1−ρ) to femtocells counterbalance decreasingspectrum access ρ_(f) to result in a higher mean femtocell throughput?

In an embodiment, given an allocation ρ_(l) at femtocell density letλ_(f), T_(f,l) and ρ_(f,l) be the mean subchannel throughput and theoptimal F-ALOHA access respectively. On increasing λ_(f) by δλ_(f) withallocation ρ_(h), let the corresponding quantities equal T_(f,h) andρ_(f,h). The femtocell network can be defined as fully-utilized [resp.sub-utilized] if a marginal increment in the femtocell density reduces[resp. improves] the mean throughput per femtocell as given below:

$\begin{matrix}{\left. {\left( {1 - \rho_{l}} \right)\rho_{f,l}T_{f,l}\begin{matrix} > \\ < \end{matrix}\left( {1 - \rho_{h}} \right)\rho_{f,h}T_{f,h}}\Leftrightarrow{\left( {1 - \rho_{l}} \right)\frac{{ASE}_{f,l}}{\lambda_{f}}\begin{matrix} > \\ < \end{matrix}\left( {1 - \rho_{h}} \right)\frac{{ASE}_{f,h}}{\lambda_{f} + {\delta \; \lambda_{f}}}}\Leftrightarrow{\frac{1 - \rho_{h}}{1 - \rho_{l}}\begin{matrix} > \\ < \end{matrix}{\frac{{ASE}_{f,l}}{{ASE}_{f,h}} \cdot \frac{\lambda_{f} + {\delta \; \lambda_{f}}}{\lambda_{f}}}} \right. = {\frac{T_{f,l}}{T_{f,h}}{\frac{\rho_{f,l}}{\rho_{f,h}}.}}} & (53)\end{matrix}$

Equation (53) reflects the competing effects of increasing allocation(1−ρ) and decreasing F-ALOHA access ρ_(f) (or increasing λ_(f)) indetermining the net femtocell throughput.

IV. NUMERICAL RESULTS

The following provides the results of a simulation of embodiments of thepresent invention. Results are presented in FIGS. 19 through 22 with thesystem parameters in FIG. 23. The number of users in each tier iscontrolled by varying N_(f). To model varying data-rate requirementsinside femtocells relative to the central macrocell, QoS values of η=0.5(equal per-user throughputs in each tier) and η=0.01 (favoring 100×higher throughput/femtocell user relative to macrocell users) areconsidered. Two propagation scenarios are presented namely 1) HighAttenuation (HA) of neighboring femtocell transmissions—modeling lowinterference—with parameters α_(f)=4 and P_(f,dB)=10 and 2) LowAttenuation (LA) scenario by setting α_(f)=3.5 and P_(f,dB)=2.

FIG. 19 shows the allocation using (19) with RR scheduling per-tier forvarying QoS parameter η. With equal average per-user throughputs(η=0.5), nearly 90% of the overall bandwidth is assigned to themacrocell. The central macrocell serves a higher number of users, whoexperience relatively poor reception. Equalizing per-user throughputsconsequently requires a significantly higher allocation to themacrocell. As η decreases, femtocells require more spectrum forproviding greater indoor capacity; e.g. in a LA scenario with η=0.01 andN_(f)=50 femtocells/cell site, nearly 70% of spectrum is allocated tofemtocells.

FIG. 20 plots the ASEs of the two-tier network using (16) for varyingQoS parameter η. In a LA scenario with η=0.01, the high degree ofco-channel interference results in the ASE maximized with fewer thanN_(f)=50 femtocells. Following Remark 2, this indicates that adding morefemtocells does not provide additional spatial reuse. In all othercases, the ASEs monotonically increase with N_(f) indicating increasingspatial reuse with addition of femtocells. To show benefits ofopportunistic scheduling, a PF scheduler provides nearly 2.3× [resp.1.35×] ASE gains relative to a RR scheduler in a HA scenario with QoSparameter η=0.5 [resp. η=0.01] and N_(f)=110 femtocells/cell site.

FIG. 21 plots the expected throughput per femtocell (1−ρ)ρ_(f)T_(f) as afunction of N_(f) and η. For η=0.5, the throughputs monotonicallyincrease with N_(f) indicating that increasing spectrum allocation (1−ρ)counteracts the effects of decreasing ρ_(f)T_(f), in effect, thefemtocell network is sub-utilized. With η=0.01 in a LA environmenthowever, the femtocell throughputs decrease with increasing N_(f),indicating that the femtocell network is fully-utilized.

FIG. 22 plots the minimum required spectrum WF, which satisfies a targetaverage data rate of D_(c)=0.1 Mbps for each macrocell user, given RRand PF scheduling at the macrocell. For each femtocell, corresponding toQoS parameter values η=0.5 and η=0.01, target average data ratesD_(f)=D_(c)(1−η)/η equaling 0.1 and 10 Mbps/user are considered. SinceProposition 1 ensures that the QoS constraint in (17) is binding, therequired spectrum WF is given as:

$\begin{matrix}{{WF} = {{\frac{U_{c}}{\rho \; T_{c}}D_{c}} = {\frac{U_{f}}{\left( {1 - \rho} \right)\rho_{f}T_{f}}D_{f}}}} & (54)\end{matrix}$

Two key observations are: First, a channel aware macrocell may providesignificant savings in the spectrum necessary to meet D_(f) and D_(c);e.g., with η=0.01 and N_(f)=50 femtocells/cell site in a HA scenario,the spectrum reduction is nearly 50% (10 MHz). Next, spatial reuse andspectrum requirements with the addition of femtocells may be markedlydifferent depending on attenuation from neighboring femtocells. Forexample, in a LA [resp. HA] scenario, the spectrum requirement WFincreases [resp. decreases] with increasing hotspot density indicatingthe femtocell network is fully-utilized [resp. sub-utilized] with theper-tier spectrum allocation in equation (19).

V. CONCLUSIONS

Embodiments of the present invention propose a decentralized spectrumallocation strategy as an alternative to centralized/coordinatedfrequency assignment in a two-tier network. The proposed allocation maydepend on the per-tier throughputs, the loading of users in each tierand the QoS requirements, accounting for co-channel interference andpath-losses. With a randomized spectrum access strategy, femtocellsshould, according to one embodiment, access a decreasing fraction oftheir allocated spectrum with increasing femtocell density, in order tomaximize spatial reuse. Spatial reuse benefits derived from channelaware macrocell scheduling result in nearly 50% spectrum reduction formeeting target per-tier data rates. In a low interference scenario whereaddition of hotspots provides increased spatial reuse, the spectrumrequirement is unchanged up to 110 femtocells/cell site. On the otherhand, the limited spatial reuse in high interference scenarios maynecessitate increasing spectrum with addition of femtocells. Theseinsights provide guidelines on performance of decentralized spectrumallocation in the two-tier networks.

Techniques for Reducing Interference in a Two-Tier MIMO Cellular Network

Another embodiment of the invention proposes use of multipleantennas—also called Multiple-Input-Multiple-Output (MIMO)transmission—at the central macrocell/femtocells. As noted above, MIMOprovides extra degrees of freedom (spatial diversity) to combatcross-tier interference in a two-tier network with universal frequencyreuse. Specifically, using multiple antennas may enable a femtocell toeither (a) eliminate cross-tier interference to nearby cellular users ina closed-access system by nulling their beams in the direction of anearby cellular user, or (b) handoff nearby cellular users and boosttheir signal strength through array processing.

With MIMO, another embodiment of the invention may enable a group ofgeographically close femtocells—to reduce communication latency—tocombine their antenna resources for collaboratively canceling cross-tierinterference to nearby cellular user(s). So, even if a single femtocellcannot cancel its interference because it does not have sufficientantennas (i.e., degrees of freedom), if there are too many cellularusers in its vicinity, this technique may overcome this difficulty asthe collaborating femtocells possess extra degrees of freedom toeliminate cross-tier interference. This technique may require thecollaborating femtocells to exchange their respective channelinformation (CI) with a nearby cellular user over the backhaul network.

With MIMO, yet another embodiment of the invention may provide anadaptive power control technique (with closed-access) and a handoffpolicy to femtocells (with open access) in a two-tier MIMO cellularsystem with universal frequency reuse. With closed access, thistechnique may enable femtocells to determine their maximum transmitpower in order to guarantee a certain minimum Quality-of-Service (QoS)to a nearby cellular user. The transmit power may be derived as afunction of the number of antennas at the macrocell/femtocells, fixedand variable components of indoor and outdoor path-losses which ariseduring terrestrial propagation of wireless signals, and fast fadingchannel scenarios (e.g. Rayleigh fading). In open access systems, thesame embodiment may be used to determine the locations of cellular userswith respect to their central macrocell, wherein a nearby femtocell mayprovide the outdoor cellular user with better QoS (through a handoff ofthe cellular user to the nearby femtocell).

Yet another embodiment of the invention with MIMO may provide a policyto handoff an indoor user from its femtocell to the central macrocell asa function of the number of antennas at the macrocell/femtocells, fixedand variable components of indoor and outdoor path-losses which ariseduring terrestrial propagation of wireless signals, and fast fadingchannel scenarios (e.g. Rayleigh fading). Such a scenario may arisebecause a femtocell which is located geographically close to the centralmacrocell may experience unacceptable cross-tier interference;consequently an indoor user may receive better Quality-of-Service bycommunicating with the central macrocell.

The following provides a more detailed description of this embodiment ofthe present invention.

Interference Mitigation Using Spatial Diversity in Two-Tier Networks I.Motivation

In a co-channel deployment of femtocells (also known as a femtocellunderlay), cross-tier interference may become a capacity-limitingfactor. With closed access, a cellular user (communicating with thecentral macrocell) in the vicinity of an active femtocell may sufferunacceptable interference. The ensuing “dead zone” around the femtocell,which is illustrated in FIG. 24, can result in poor outdoor cellularreception.

Embodiments of the present invention propose two prior approaches totackle cross-tier interference in a shared spectrum two-tier networknamely 1) interference avoidance and 2) interference aware power controlschemes at each femtocell. Employing multiple antennas at a femtocellbase station (BS) can offer a third possibility namely to utilize theavailable spatial degrees of freedom at the femtocell to serve multipleindoor users and eliminate interference at outdoor cellular users.

According to one embodiment, multiple antennas at femtocells can be usedto serve multiple indoor users—providing multiplexing gain, andeliminating cross-tier interference by nulling transmissions in thedirection of a nearby cellular user. Consequently, an outdoor cellularuser can obtain reliable reception even though it is in the vicinity ofan active femtocell. Additionally, indoor users can continue toexperience the superior reception which was a primary reason why theyinstalled a femtocell in the first place.

Since a femtocell may have only a limited number of antennas,sacrificing these degrees of freedom for interference cancellation mayresult in a tradeoff due to smaller multiplexing gain. On the otherhand, the co-channel interference at a nearby cellular user may besevere enough to force femtocells to expend resources to tackle theinterference issue. The questions to be addressed are as follows:

1) Given a performance metric which is the weighted sum of theachievable rates provided by the macrocell and femtocell respectively,to how many cellular users should a femtocell cancel its interference?Alternatively, how many users should a femtocell serve in each signalinginterval? How does this depend on the path losses between the femtocelland the macrocell to the cellular user?

2) With a small number of transmit antenna per femtocell, the availabledegrees of freedom at each femtocell may not be sufficient to performinterference cancellation. How can multiple femtocells “group” tocooperatively cancel their interference to nearby cellular users?

II. System Model

For simplicity, assume a single central macrocell B₀ with T_(c) antennasand a single femtocell B₁ with T_(f) antennas. Assume that B₀ transmitsto a set S_(c) consisting of S_(c) users (1≦S_(c)≦T_(c)) users/slot. Asillustrated in FIG. 25 (in which S_(f) refers to the number of activeusers in femtocell, s_(f) is the data symbol for indoor user f, wrepresents the beamforming vector per femtocell user, and F representsthe linear precanceler to null interference to cellular user), thefemtocell transmits to a set S_(f) consisting of S_(f) users/slot where1≦S_(f)≦T_(f). Here the term “slot” can refer to atime/frequency/orthogonal code resource.

Assume femtocell B₁ cancels its interference to a subset S_(c,f) ⊂S_(c)consisting of S_(c,f)=|S_(c,f)| cellular users. Because of thedimensionality constraint, the number of scheduled indoor users S_(f)can be no more than T_(f)−S_(c,f).

Assumption 1: Perfect channel information is assumed at the femtocellregarding its channels to users in S_(f) as well as S_(c,f).

Designate the matrix Q^(†)=[g₁ g₂ . . . g_(S) _(c,f) ] to consist of thevector channels between B₁ and each user in S_(f). For cancelinginterference, B₁ designs a precoder F which lies in the T_(f)−|S_(f)|dimensional null space of Qε

S_(c,f)×T_(f). For example, if S_(f)=1 (one user per femtocell), theprecoder F can be chosen to be the orthogonal projector

${I - {\frac{g_{1}g_{1}^{\dagger}}{g}.}}\;$

In the general case, one chooses Fε

T_(f)×(S−S_(c,f)) to equal the T_(f)−S_(c,f) right singular vectors inthe full SVD decomposition of Qε

S_(c,f)×T_(f).

Assume that B₁ serves multiple indoor users by designing unit-normbeamforming vectors {w_(i)} which lie in the null space of Q. Inaddition, the symbol vector s consisting of the information symbols tothe S_(f) indoor users satisfies the individual power constraint

[∥s∥²]=P_(f). Ignoring cross-tier interference from the macrocell B₀ ateach indoor user, the received signal model for a femtocell user can berepresented as:

$\begin{matrix}{{y_{i} = {{\sqrt{\gamma}g_{i}^{\dagger}F{\sum\limits_{j = 1}^{S_{f}}{w_{j}s_{j}}}} + n_{i}}},{\forall{i \in S_{c,f}}}} & (55)\end{matrix}$

where γ is the channel power from the femtocell to indoor users.

Assuming S_(f)≦T_(f)−S_(c,f) active indoor users at femtocell B₁, theergodic sum rate provided by B₁ is given as:

$\begin{matrix}{{R_{Femto}\left\lbrack S_{f} \right\rbrack} = {S_{f} \cdot {\left\lbrack {\log_{2}\left( {1 + \frac{\gamma \frac{P_{f}}{S_{f}}{{w_{1}^{\dagger}g_{1}}}^{2}}{{\sum\limits_{j \neq i}{\gamma \frac{P_{f}}{S_{f}}{{w_{j}^{\dagger}g_{1}}}^{2}}} + \sigma^{2}}} \right)} \right\rbrack}}} & (56)\end{matrix}$

As seen in equation (56), setting S_(c,f)=S_(f) can allow a femtocell toachieve its full multiplexing gain but can result in maximum cross-tierinterference to cellular users. Conversely, setting S_(c,f)=1 allows afemtocell to cancel cross-tier interference to S_(f)−1 cellular users,but can allow a multiplexing gain of just 1.

From the macrocell B₀'s perspective, there are two sets of cellularusers namely 1) the |S_(c,f)| cellular users to which B₁ cancels itsinterference, and 2) the remaining |S\S_(c,f)| cellular users whocommunicate with B₀ but experience cross-tier interference (given twosets A and B⊂A, the notation A\B denotes the set A∩_(B)C).

Assumption 2: All cellular users experience identical channel powers αand β to the macrocell B₀ and femtocell B₁ respectively.

Assumption 2 can be justified if one assumes that users in S_(c,f) aresituated in the vicinity of a femtocell (see FIG. 26 for example whereinD_(c) represents the distance of cellular user to macrocell, and D_(f)represents the distance of cellular user to femtocell), or alternativelyassuming that these users are co-located from each BS's perspective.

Designate the matrix H^(†)=[h₁ h₂ . . . h_(S) _(c) ] consisting of the(T_(c)×1) vector channels between the central macrocell B₀ and eachcellular user in S_(c). Assume that B₀ employs a set of beamformingvectors {v_(j)},∥v_(j)∥=1 to serve multiple cellular users. Withzero-forcing beamforming (ZFBF) for example, {v_(j)} are chosen as thenormalized columns of the matrix H^(†)(HH^(†))⁻¹. The received signalfor a cellular user in S_(c,f) [resp. S\S_(c,f)] can be given as:

$\begin{matrix}{{y_{i} = {{\sqrt{\alpha}h_{i}^{\dagger}{\sum\limits_{j = 1}^{S_{c}}{v_{j}s_{j}}}} + {n_{i}{\forall{i \in S_{c,f}}}}}}{y_{i} = {{\sqrt{\alpha}h_{i}^{\dagger}{\sum\limits_{j = 1}^{S_{c} - S_{c,f}}{v_{j}s_{j}}}} + {\sqrt{\beta}g_{i}^{\dagger}F{\sum\limits_{k = 1}^{S_{f}}{w_{k}s_{k}}}} + {n_{i}{\forall{i \in {S/S_{c,f}}}}}}}} & (57)\end{matrix}$

For a given set of channel powers α and β and a representative user 1 inS_(c,f) [resp. 2 in S_(c)\S_(c,f)] and assuming equal power allocation,the ergodic sum rate for the macrocell can be given as:

$\begin{matrix}\begin{matrix}{R_{Macro} = {{S_{c,f} \cdot {\left\lbrack {\log_{2}\left( {1 + \frac{\frac{\alpha \; P_{c}}{S_{c}}{{v_{i}^{\dagger}h_{1}}}^{2}}{{\sum\limits_{j \neq 1}{\frac{\alpha \; P_{c}}{S_{c}}{{v_{j}^{\dagger}h_{1}}}^{2}}} + \sigma^{2}}} \right)} \right\rbrack}} +}} \\{{\left( {S_{c} - S_{c,f}} \right) \cdot {\left\lbrack {\log_{2}\left( {1 + \frac{\frac{\alpha \; P_{c}}{S_{c}}{{v_{i}^{\dagger}h_{2}}}^{2}}{\begin{matrix}{{\sum\limits_{j \neq i}{\frac{\alpha \; P_{c}}{S_{c}}{{v_{j}^{\dagger}h_{2}}}^{2}}} +} \\{{\sum\limits_{l = 1}^{S_{f}}{\frac{\beta \; P_{f}}{S_{f}}{{w_{i}^{\dagger}{Fg}_{2}}}^{2}}} + \sigma^{2}}\end{matrix}}} \right)} \right\rbrack}}} \\{= {{R_{Macro}\left( _{c,f} \right)} + {R_{Macro}\left( {_{r}\backslash _{c,f}} \right)}}}\end{matrix} & (58)\end{matrix}$

where R_(Macro) (S_(c,f)) and R_(Macro) (S\S_(c,f)) designate theergodic sum rates obtained by cellular users in S_(c,f) and S\S_(c,f)respectively.

A. Served Users Per Femtocell to Maximize Weighted Ergodic Sum Rate

One objective is to determine the optimal number of femtocell users thatshould be scheduled in order to maximize the weighted ergodic sum ratesprovided to indoor users and cellular users in Sc,f. Allocate weights θ,0≦θ≦1 and 1−θ to the femtocell and macrocell rates in equations (56) and(58) respectively. Then, maximizing the convex combination of theper-tier rates results in the Pareto-frontier of the system. Onequestion posed herein is: Given path-losses α and β and a weightingparameter θ, how many users should a femtocell serve? According toembodiments of the present invention, the number of served indoor usersmay be given as the solution to the following optimization problem:

$\begin{matrix}{{{\overset{\sim}{S}}_{c,f}(\theta)} = {{\underset{1 \leq S_{f} \leq T_{f}}{argmax}\theta \; {R_{Femto}\left\lbrack S_{c,f} \right\rbrack}} + {\left( {1 - \theta} \right){R_{Macro}\left( _{c,f} \right)}}}} & (59)\end{matrix}$

In (59), the quantity {tilde over (S)}_(f)(θ) is a function of thechannel powers parameterized by α and β, respectively.B. Maximize Instantaneous Weighted Sum Rate with User Selection andLimited Feedback

The scenario considered here is that a femtocell decides a certain setof indoor users S_(f) and a set of cellular users S_(c,f), in order tomaximize the instantaneous sum rate. The user selection may be conductedeither with full CSIT at the femtocell, or assuming limited feedbackfrom indoor users to the femtocell using Random Vector Quantization(RVQ) based code-books.

One idea here is that a femtocell should select the cellular users forinterference cancellation and indoor users for multiplexing, such thatthe overall weighted sum rate can be maximized. Mathematically, thisselection procedure can be given as the optimization problem:

$\begin{matrix}{\left( {{\overset{\sim}{}}_{c,f},{\overset{\sim}{}}_{f}} \right) = {{{argmax}\; {\theta \cdot {\sum\limits_{i \in _{f}}{\log_{2}\left( {1 + \frac{\gamma \frac{P_{f}}{S_{f}}{{w_{j}^{\dagger}g_{j}}}^{2}}{{\sum\limits_{{j \neq i},{j \in _{f}}}{\gamma \frac{P_{f}}{S_{f}}{{w_{j}^{\dagger}g_{i}}}^{2}}} + \sigma^{2}}} \right)}}}} + {\left( {1 - \theta} \right) \cdot {\sum\limits_{l \in _{c,f}}{\log_{2}\left( {1 + \frac{\alpha \frac{P_{c}}{S_{f}}{{v_{l}^{\dagger}h_{l}}}^{2}}{{\sum\limits_{{k \neq l},{k \in _{c}}}{\frac{\alpha \; P_{c}}{S_{c}}{{v_{k}^{\dagger}h_{l}}}^{2}}} + \sigma^{2}}} \right)}}}}} & (60)\end{matrix}$

III. Distributed MIMO by Femtocell Coordination Across Backhaul

In this final embodiment, consider the scenario when there are morecellular users in the vicinity of each femtocell relative to theavailable degrees of freedom available to each femtocell. In such ascenario, embodiments of the present invention propose that neighboringfemtocells communicate (possibly with a centralized entity over theinternet backhaul) and use distributed MIMO to form a “meta-femtocell”which can cancel interference to both cellular users simultaneously.

For example, with 2 antennas per femtocell and 1 active indoor user perfemtocell, each femtocell can cancel interference to just 1 cellularuser. By jointly designing the precoder F, resulting 4 antenna BTS cancancel interference to both cellular users simultaneously (see FIG. 27).More generally, deploying more femtocells may strategically help anoperator reduce interference caused by a rogue femtocell, by inducingthat femtocell to coordinate with other femtocells in its immediatevicinity. Perfect synchronization is assumed here which, along with IPlatency, may be one of the key issues facing a practical implementation.

Outage Probability and Threshold Distances in Two-Tier Networks withSpatial Diversity

I. Motivation

As noted above, in a co-channel deployment of femtocells (also known asa femtocell underlay), cross-tier interference may become acapacity-limiting factor. With closed access, a cellular user(communicating with the central macrocell) in the vicinity of an activefemtocell may suffer unacceptable interference. The ensuing “dead zone”(FIG. 24) around the femtocell can result in poor outdoor cellularreception.

This embodiment of the present invention considers a scenario whereinthe central macrocell and the femtocells share a common region ofspectrum and transmit using multiple antennas to their respective users.Femtocells are assumed to be randomly located on the plane according toa Spatial Poisson Point Process (SPPP) Λ_(f) with intensity λ_(f) whichdescribes the mean number of femtocells per unit area. Assuming perfectchannel state information (CSIT) at each base station (BS) to itsrespective users and assuming that each BS transmits using beamformingto a single user (SUBF), this embodiment contains at least the followingtwo contributions:

1) The locations of cellular user with respect to its central macrocellbeyond which a certain minimum Quality-of-Service (QoS) requirement(outage probability) likely cannot be satisfied because of excessivecross-tier interference from cochannel femtocells.

2) The locations of a femtocell with respect to the central macrocellwherein cross-tier interference from the macrocell may be significantenough to cause outage at the femtocell.

The above expressions may be used in practical deployments of a two-tiernetwork to either perform adaptive power control at femtocells inclosed-access schemes, or perform handoff from the central macrocell toa nearby femtocell in a open-access scheme.

II. System Model

According to one embodiment, for simplicity, assume a single centralmacrocell B₀ using M antennas to service a geographical region C. Themacrocell is underlaid with randomly located femtocells distributedaccording to a SPPP Λ_(f) of intensity λ_(f). Consequently, in theregion C with area |C|, there are an average of N_(f)=λ_(f)|C| cochannelfemtocells per cell site. In addition, each femtocell is assumed to haveF antennas. All cellular and indoor femtocell users are assumed to haveexactly one antenna at their handset.

Each BS transmits to exactly one user per slot—WLOG, the slot may referto either a time or frequency or an orthogonal code resource—fortransmission by beamforming (abbreviated as SUBF) in the direction ofthe vector channel to its user.

Assumption 1: Perfect channel information is assumed at the centralmacrocell [resp. femtocells] regarding the channels to their individualusers.

The terrestrial propagation losses between each BS to surrounding userscan be modeled using the IMT-2000 channel model. Embodiments of thepresent invention consider the following scenarios:

Macrocell to Cellular Users. The path loss between B₀ and cellular userscan be modeled as:

$\begin{matrix}{{PL}_{C,{d\; B}} = {{10\; \alpha_{c}{\log_{10}(D)}} + {30\; \log_{10}f_{c}} - 71}} & (61) \\{\mspace{76mu} {= {A_{c,{d\; B}} + {10\; \alpha_{c}\log_{10}D}}}} & (62)\end{matrix}$

-   -   where        -   PL_(C),_(dB)=Path losses between B₀ to cellular user        -   α_(c)=Outdoor path-loss exponent        -   D=Distance of cellular user to B₀        -   f_(c)=Carrier frequency in MHz

Macrocell to Indoor Femtocell Users. The path loss between the centralmacrocell to indoor femtocell users can be modeled as:

$\begin{matrix}{{PL}_{F,C,{d\; B}} = {{10\; \alpha_{c}{\log_{10}(D)}} + {30\log_{10}f_{c}} - 71 + P_{d\; B}}} & (63) \\{ {= {A_{f,c,{d\; B}} + {10\; \alpha_{c}{\log_{10}(D)}}}}} & (64)\end{matrix}$

-   -   where PL_(F),_(C,dB)=Path losses between B₀ to indoor user        -   P_(dB)=Wall partition loss to indoor user

Femtocell to Outdoor Cellular Users. The path loss between a femtocellto a cellular user can be modeled as:

$\begin{matrix}{{PL}_{C,F,{d\; B}} = {37 + {10\; \alpha_{fo}{\log_{10}(D)}} + P_{d\; B}}} & (65) \\{ {= {A_{c,f,{d\; B}} + {10\; \alpha_{fo}{\log_{10}(D)}}}}} & (66)\end{matrix}$

-   -   where PL_(C),_(F,dB)=Path losses between femtocell to a cellular        user        -   α_(fo)=Indoor to outdoor path-loss exponent            Path-loss between a given femtocell to a neighboring            femtocell can be obtained by adding an extra P_(dB) term to            model the double wall partition losses during            indoor-to-indoor propagation.

Femtocell to In-home Users. The path loss between a femtocell to indoorusers can be modeled as:

$\begin{matrix}{{PL}_{F,F,{d\; B}} = {37 + {10\; \alpha_{fi}{\log_{10}\left( R_{F} \right)}}}} & (67) \\{= {A_{{fi},{d\; B}} + {10\; \alpha_{fi}{\log_{10}(D)}}}} & (68)\end{matrix}$

where

-   -   PL_(F),_(F,dB)=Path losses between a femtocell to its indoor        users    -   R_(F)=Radio range of a femtocell    -   α_(fi)=Indoor path-loss exponent

A. Outage Probability for a Cellular User

Assume that the macrocell B₀ selects user 0 for transmission. Thebeamforming vector w₀ can be chosen to be unit norm and in the directionof the vector downlink channel h₀ whose entries are distributed as CN(0,1). With the assumption of perfect CSIT,

$w_{0} = {\frac{h_{0}}{h_{0}}.}$

The received signal for a cellular user 0 at distance D to the centralmacrocell B₀ can be written as:

$\begin{matrix}{y_{0} = {{\sqrt{A_{c}}D^{{- \alpha_{c}}/2}h_{0}^{\dagger}w_{0}s_{0}} + {\sum\limits_{i \in \Lambda_{f}}{\sqrt{A_{c,f}}{X_{0,i}}^{{- \alpha_{fo}}/2}e_{0,i}^{\dagger}w_{i}r_{i}}}}} & (69)\end{matrix}$

where s₀ refers to the data symbol intended for cellular user 0 with

[|s₀|²]=P_(c). Similarly, r_(i)∀iεΛ_(f) refers to the data symbol forthe scheduled user in femtocell F_(i) with

[|r_(i)|²]=P_(f).

Assuming that all receivers are interference limited (as may be the casein a well-designed cellular network), the Signal-to-Interference (SIR)ratio for the cellular user 0 can be given as:

$\begin{matrix}{{{SIR}_{C,{{CSIT}\text{-}{SUBF}}}(D)} = \frac{A_{c}P_{c}{h_{0}}^{2}}{\sum\limits_{i \in \Lambda_{f}}{A_{c,f}P_{f}{{e_{0,i}^{\dagger}v_{i}}}^{2}{X_{0,i}}^{- \alpha_{fo}}}}} & (70)\end{matrix}$

Assuming a target SIR equaling β, a Quality-of-Service (QoS) requirementfor user 0 stipulates that the target SIR is satisfied with aprobability of at least 1−ε. In the small ε regime, one can derive theoutage probability using (62) which can be given as:

$\begin{matrix}\begin{matrix}{{{\mathbb{P}}_{out}\left( {\beta,D} \right)} = {\Pr \left( {{{SIR}_{C,{{CSIT}\text{-}{SUBF}}}(D)} \leq \beta} \right)}} \\{= {\lambda_{f}{C\left( {\beta \; Q_{D}} \right)}^{2/\alpha_{fo}}\left( {1 + {\sum\limits_{k = 0}^{M - 1}\frac{\prod\limits_{l = 0}^{k - 1}\; \left( {l - {2/\alpha_{fo}}} \right)}{k!}}} \right)}}\end{matrix} & (71) \\{where} & \; \\{{Q_{D} = {\frac{A_{f,c}}{A_{c}}\frac{P_{f}}{P_{c}}}}\mspace{14mu} {{{and}\mspace{14mu} C} = {\frac{2\pi}{\alpha_{fo}}{\Gamma \left( {2/\alpha_{fo}} \right)}{\Gamma \left( {1 - {2/\alpha_{fo}}} \right)}\mspace{14mu} {{respectively}.}}}} & \;\end{matrix}$

Consequently, in order to satisfy the QoS requirement of ε for a givenintensity of interfering femtocells λ_(f), one can obtain the followingcondition for D:

$\begin{matrix}{\left. {{{\mathbb{P}}_{out}\left( {\beta,D} \right)} \leq \varepsilon}\Rightarrow{D \leq D_{\max}} \right. = {\left\lbrack {\frac{A_{c}}{A_{c,f}}\frac{P_{c}}{P_{f}}\varepsilon^{\alpha_{fo}/2}} \right\rbrack^{1/\alpha_{c}} \cdot \left\{ {\lambda_{f}C\; {\beta^{2/\alpha_{fo}}\left\lbrack {1 + {\sum\limits_{k = 0}^{M - 1}\frac{\prod\limits_{l = 0}^{k - 1}\; \left( {l - {2/\alpha_{fo}}} \right)}{k!}}} \right\rbrack}} \right\}^{\frac{- \alpha_{fo}}{2\alpha_{c}}}}} & (72)\end{matrix}$

Given M antennas at the macrocell, a target SIR β, a QoS requirement εand an average of N_(f)=λ_(f)|C| femtocells per cell site, (72) canprovide the threshold distance of a cellular user to its macrocellD_(max) beyond which the cellular user suffers unacceptable cross-tierinterference.

The foregoing results may motivate the following two embodiments of thepresent invention:

Handoff with Open Access Scheme. Hand off all cellular users located atD>D_(max) to a nearby femtocell.

Power control with Closed Access Scheme. Given a cellular user atdistance D to its BS, use (72) to determine the transmit power P_(f) forneighboring femtocells, in order that user 0 obtains an SIR of β with aprobability greater than 1−ε.

B. Outage Probability of a Femtocell User

In this section, a reference femtocell F₀ at distance D from the centralmacrocell B₀ is considered. The outage probability at F₀ may be computedconsidering only the cross-tier interference from B₀. The signalreceived at user 0 inside femtocell F₀ can be given as:

y ₀ =√{square root over (A)} _(fi) R _(f) ^(α) ^(fi) ^(/2) g ₀ ^(†) w ₀r ₀+√{square root over (A _(f,c))}{dot over (D)} ^(−α) ^(c) ^(/2) f _(c)^(†) w ₀ s ₀  (73)

where g₀ [resp. f_(c)] are the vector downlink channels from femtocellF₀ [resp. macrocell B₀].

The received SIR for the in-home user 0 from the transmitted signal atF₀ is upper bounded as:

$\begin{matrix}{{{SIR}_{F,{{CSIT}\text{-}{SUBP}}}(D)} \leq \frac{P_{f}R_{f}^{{- \alpha_{fi}}/2}{g_{0}}^{2}}{P_{c}D^{- \alpha_{c}}{{f_{c}^{\dagger}w_{0}}}^{2}}} & (74)\end{matrix}$

With a target SIR β for the femtocell user, the outage probabilityP_(out)(β, D)=Pr(SIR_(F),_(CSIT−SUBT)(D)≦β) can be given as:

$\begin{matrix}{\begin{matrix}{{{\mathbb{P}}_{out}\left( {\beta,D} \right)} = {\Pr\left( {{{SIR}_{F,\underset{F}{{CSIT}\text{-}{SUBT}}}(D)} \leq \beta} \right)}} \\{= \frac{\kappa \left( {\beta,D} \right)}{1 + {\kappa \left( {\beta,D} \right)}}}\end{matrix}{where}{{\kappa \left( {\beta,D} \right)} = {\frac{P_{c}}{P_{f}}\frac{D^{- \alpha_{c}}}{R_{f}^{- \alpha_{fi}}}\frac{A_{f,c}}{A_{fi}}\beta}}} & (75)\end{matrix}$

Consequently, a QoS requirement ε which stipulates that the outageprobability for a femtocell user is no more than ε is violated wheneverthe femtocell is less than a distance D_(min) given as:

$\begin{matrix}{{D \leq D_{\min}} = \left. {\left\lbrack {\frac{A_{f,c}}{A_{fi}}\frac{P_{c}}{P_{f}}R_{f}^{1/\alpha_{c}}\beta} \right\rbrack^{1/\alpha_{c}}\left( \frac{\varepsilon^{1/F}}{1 - \varepsilon^{1/F}} \right)^{{- 1}/\alpha_{c}}}\Rightarrow{{{\mathbb{P}}_{out}\left( {\beta,D} \right)} > \varepsilon} \right.} & (76)\end{matrix}$

The above expression can provide the threshold distance of a femtocellwith respect to the central macrocell B₀ within which cross-tierinterference prevents in-home users from satisfying their QoS constraintε.

Consequently, embodiments of the present invention propose that allin-home users located at D≦D_(min) communicate with the centralmacrocell.

III. Simulation Results

FIGS. 28, 29 and 30 illustrate simulation results associated with theforegoing embodiments of the present invention. In particular, FIG. 28provides the system parameters used for the simulations; FIG. 29illustrates the outage probability at a femtocell as a function of itsdistance from the central macrocell; and FIG. 30 illustrates thethreshold distance of cellular user at which QoS requirement ε=0.1 isviolated as a function of the number of femtocells per cell site and thenumber of transmit antennas at the macrocell M.

CONCLUSION

While the methods and systems have been described in connection withembodiments and specific examples, it is not intended that the scope belimited to the particular embodiments set forth, as the embodimentsherein are intended in all respects to be illustrative rather thanrestrictive

The examples described herein are put forth so as to provide those ofordinary skill in the art with a complete disclosure and description ofhow the system and/or methods claimed herein are designed and evaluated,and are intended to be purely exemplary and are not intended to limitthe scope of the methods and systems.

Unless otherwise expressly stated, it is in no way intended that anymethod set forth herein be construed as requiring that its steps beperformed in a specific order. Accordingly, where a method claim doesnot actually recite an order to be followed by its steps or it is nototherwise specifically stated in the claims or descriptions that thesteps are to be limited to a specific order, it is no way intended thatan order be inferred, in any respect. This holds for any possiblenon-express basis for interpretation, including: matters of logic withrespect to arrangement of steps or operational flow; plain meaningderived from grammatical organization or punctuation; the number or typeof embodiments described in the specification.

As described above and as will be appreciated by one skilled in the art,embodiments of the present invention may be configured as a system,method or network entity. Accordingly, embodiments of the presentinvention may be comprised of various means including entirely ofhardware, entirely of software, or any combination of software andhardware. Furthermore, embodiments of the present invention may take theform of a computer program product on a computer-readable storage mediumhaving computer-readable program instructions (e.g., computer software)embodied in the storage medium. Any suitable computer-readable storagemedium may be utilized including hard disks, CD-ROMs, optical storagedevices, or magnetic storage devices.

Embodiments of the present invention have been described above withreference to block diagrams and flowchart illustrations of methods,apparatuses (i.e., systems) and computer program products. It will beunderstood that each block of the block diagrams and flowchartillustrations, and combinations of blocks in the block diagrams andflowchart illustrations, respectively, can be implemented by variousmeans including computer program instructions. These computer programinstructions may be loaded onto a general purpose computer, specialpurpose computer, or other programmable data processing apparatus, suchas processor 103 discussed above with reference to FIG. 3, to produce amachine, such that the instructions which execute on the computer orother programmable data processing apparatus create a means forimplementing the functions specified in the flowchart block or blocks.

These computer program instructions may also be stored in acomputer-readable memory that can direct a computer or otherprogrammable data processing apparatus (e.g., processor 103 of FIG. 3)to function in a particular manner, such that the instructions stored inthe computer-readable memory produce an article of manufacture includingcomputer-readable instructions for implementing the function specifiedin the flowchart block or blocks. The computer program instructions mayalso be loaded onto a computer or other programmable data processingapparatus to cause a series of operational steps to be performed on thecomputer or other programmable apparatus to produce acomputer-implemented process such that the instructions that execute onthe computer or other programmable apparatus provide steps forimplementing the functions specified in the flowchart block or blocks.

Accordingly, blocks of the block diagrams and flowchart illustrationssupport combinations of means for performing the specified functions,combinations of steps for performing the specified functions and programinstruction means for performing the specified functions. It will alsobe understood that each block of the block diagrams and flowchartillustrations, and combinations of blocks in the block diagrams andflowchart illustrations, can be implemented by special purposehardware-based computer systems that perform the specified functions orsteps, or combinations of special purpose hardware and computerinstructions.

Many modifications and other embodiments of the inventions set forthherein will come to mind to one skilled in the art to which theseembodiments of the invention pertain having the benefit of the teachingspresented in the foregoing descriptions and the associated drawings.Therefore, it is to be understood that the embodiments of the inventionare not to be limited to the specific embodiments disclosed and thatmodifications and other embodiments are intended to be included withinthe scope of the appended claims. Moreover, although the foregoingdescriptions and the associated drawings describe exemplary embodimentsin the context of certain exemplary combinations of elements and/orfunctions, it should be appreciated that different combinations ofelements and/or functions may be provided by alternative embodimentswithout departing from the scope of the appended claims. In this regard,for example, different combinations of elements and/or functions thanthose explicitly described above are also contemplated as may be setforth in some of the appended claims. Although specific terms areemployed herein, they are used in a generic and descriptive sense onlyand not for purposes of limitation.

1. A system comprising: one or more femtocell base stations deployedwithin a range of a cellular base station, wherein the cellular basestation utilizes a frequency band and respective femtocell base stationsutilize a frequency band substantially the same as the cellular basestation frequency band, wherein respective femtocell base stations areconfigured to employ one or more interference avoidance techniques suchthat coexistence between the cellular base station and the correspondingfemtocell base station is enabled.
 2. The system of claim 1, whereinrespective femtocell base stations employ code division multiple accesstransmission.
 3. The system of claim 2, wherein in order to reduceinterference, respective femtocell base stations further comprise one ormore directional antennas.
 4. The system of claim 2, wherein at leastone of the interference avoidance techniques comprises use of randomizedtime hopping when transmitting signals to and from one or more usersassociated with the corresponding femtocell base station.
 5. The systemof claim 2, wherein at least one of the interference avoidancetechniques comprises use of randomized frequency hopping whentransmitting signals to and from one or more users associated with thecorresponding femtocell base station.
 6. The system of claim 1, whereinrespective femtocell base stations employ frequency division multipleaccess transmission.
 7. The system of claim 6, wherein respectivefemtocell base stations perform frequency division multiple accesstransmission using Orthogonal Frequency Division Multiple Access (OFDMA)technique, wherein one or more frequency subchannels are created by aninverse Fourier transform.
 8. The system of claim 6 further comprising:a central radio access network controller associated with the cellularbase station, said central radio access network controller configured todynamically partition a frequency spectrum associated with the cellularbase station and the one or more femtocell base stations into a firstportion to be used by one or more users associated with the cellularbase station and a second portion to be used by one or more usersassociated with respective femtocell base stations.
 9. The system ofclaim 6, wherein at least one of the interference avoidance techniquescomprises randomly selecting a predetermined number of availablefrequency subchannels for transmitting signals to and from thecorresponding femtocell users.
 10. The system of claim 6, wherein atleast one of the interference avoidance techniques comprises identifyingone or more unutilized frequency subchannels for transmitting signals toand from the corresponding femtocell users.
 11. The system of claim 1,wherein at least one of the interference avoidance techniques comprisesuse of two or more transmit antennas and two or more receive antennas.12. The system of claim 11, wherein at least one of the interferenceavoidance techniques comprises nulling one or more transmissions in adirection of a nearby user associated with the cellular base station.13. The system of claim 11, wherein at least one of the interferenceavoidance techniques comprises handing off at least one user associatedwith the cellular base station to one of the femtocell base stations.14. The system of claim 11, wherein at least one of the interferenceavoidance techniques comprises handing off at least one user associatedwith at least one of the femtocell base stations to the cellular basestation.
 15. The system of claim 11, wherein at least one of theinterference avoidance techniques comprises reducing the transmissionpower of at least one of the femtocell base stations to reduceinterference and ensure a minimum quality-of-service for a user servicedby the cellular base station.
 16. The system of claim 1, whereinrespective femtocell base stations further comprise: at least oneantenna; a low-power amplifier; one or more analog filters configured toreject out of band interferences; a digital processing circuit forperforming digital modulation and demodulation and error correctioncoding and decoding; and a microprocessor.
 17. The system of claim 1,wherein respective femtocell base stations provide a range of coveragethat is shorter than a range of coverage associated with the cellularbase station.
 18. A method comprising: establishing one or morefemtocell base stations within a range of a cellular base station,wherein the cellular base station utilizes a frequency band andrespective femtocell base stations utilize a frequency bandsubstantially the same as the cellular base station frequency band; andemploying, by respective femtocell base stations, one or moreinterference avoidance techniques such that coexistence between thecellular base station and the corresponding femtocell base station isenabled.
 19. The method of claim 18, wherein the respective femtocellbase stations employ code division multiple access (CDMA) transmission.20. The method of claim 19, wherein employing one or more interferenceavoidance techniques further comprises transmitting one or more signalsfrom one or more users associated with the corresponding femtocell basestation using one or more directional antennas.
 21. The method of claim19, wherein employing one or more interference avoidance techniquesfurther comprises transmitting one or more signals from one or moreusers associated with the corresponding femtocell base station usingrandomized time hopping.
 22. The method of claim 21, whereintransmitting one or more signals using randomized time hopping furthercomprises: dividing an available time duration associated with a CDMAtransmission duration into a predetermined number of hopping slots; andrandomly selecting a hopping slot for transmission of a signal to orfrom the femtocell user.
 23. The method of claim 19, wherein employingone or more interference avoidance techniques further comprisestransmitting one or more signals to or from one or more users associatedwith the corresponding femtocell base station using randomized frequencyhopping.
 24. The method of claim 18, wherein respective femtocell basestations employ frequency division multiple access transmission.
 25. Themethod of claim 24, wherein respective femtocell base stations performfrequency division multiple access using Orthogonal Frequency DivisionMultiple Access (OFDMA), wherein one or more frequency subchannels arecreated by an inverse discrete Fourier transform.
 26. The method ofclaim 24, wherein employing one or more interference avoidancetechniques further comprises: dynamically partitioning a frequencyspectrum associated with the cellular base station and one or morefemtocell base stations into a first portion and a second portion;allocating the first portion to one or more users associated with thecellular base station; and allocating the second portion to one or moreusers associated with respective femtocell base stations.
 27. The methodof claim 26, wherein dynamically partitioning the frequency spectrumfurther comprises: determining a number of users associated with thecellular base station; determining a number of users associated witheach of the femtocell base stations; and dynamically partitioning thespectrum based on a comparison of the number of users associated withthe cellular base station and femtocell base stations.
 28. The method ofclaim 24, wherein employing one or more interference avoidancetechniques further comprises, respective femtocell base stationsrandomly selecting a predetermined number of available frequencysubchannels for transmitting signals to and from one or more usersassociated with the corresponding femtocell base station.
 29. The methodof claim 24, wherein employing one or more interference avoidancetechniques further comprises, respective femtocell base stationsidentifying one or more unutilized frequency subchannels fortransmitting signals to and from one or more users associated with thecorresponding femtocell base station.
 30. The method of claim 29,wherein identifying one or more unutilized frequency subchannels furthercomprises identifying the one or more unutilized frequency subchannelsthrough energy detection.
 31. The method of claim 18, wherein employingone or more interference avoidance techniques further comprisestransmitting and receiving one or more signals to and from one or moreusers associated with the corresponding femtocell base station using twoor more transmit antennas and two or more receive antennas,respectively.
 32. The method of claim 31, wherein at least one of theinterference avoidance techniques comprises nulling one or moretransmissions in a direction of a nearby user associated with thecellular base station.
 33. The method of claim 31, wherein at least oneof the interference avoidance techniques comprises handing off at leastone user associated with the cellular base station to one of thefemtocell base stations.
 34. The method of claim 31, wherein at leastone of the interference avoidance techniques comprises handing off atleast one user associated with at least one of the femtocell basestations to the cellular base station.
 35. The method of claim 31,wherein at least one of the interference avoidance techniques comprisesreducing the transmission power of the femtocell base stations to reduceinterference and ensure a minimum quality-of-service for a user servicedby a cellular base station.